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Statistics Chapter 4 — Transcript

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0:00Chapter 4, Probability. Introduction
0:04Welcome to Chapter 4, Statistics.
0:06Probability is the branch of mathematics
0:08that studies chance and uncertainty. It
0:11helps us predict how likely an event is
0:13to occur and forms a foundation of
0:15inferial statistics. In this chapter, we
0:18will learn how to calculate
0:20probabilities and apply them to real
0:22life situations.
2:41All right. Uh before we
2:44start, is there anything you need to ask
2:46me about uh what I covered last time?
2:53Okay, let's talk about addition rule of
2:56probability. So I give you the formula.
2:59So if you have two events
3:03with the connective word or the
3:07probability of A or B, A is an event, B
3:12is also an event. So to find the
3:14probability will be P of A plus P of B
3:18minus P of A and B. But if the two
3:23events are mutually exclusive
3:27the pos of P of A or B will be just P of
3:31A plus P of B. You do not need the minus
3:34P of A and B. All right. So let's try
3:36this uh example here.
3:43Okay,
3:46let's try this one.
3:52>> Can I see it again?
3:56>> Hello.
3:56>> So, I can take a picture.
3:59>> Are you Are you talking to me?
4:01>> Yes.
4:02>> What did you say?
4:04>> Um, can I see the slide? The first one,
4:07please. I think uh you get this uh last
4:10time I asked you to copy that last time.
4:14Check your notes.
4:16>> Okay.
4:17>> Okay.
4:18>> Okay.
4:19>> Okay. Good. Now,
4:20>> yes.
4:22>> Let's try this one here.
4:26Solution to the blue. Yeah.
4:30Okay. So, if a single die Okay. So
4:35talking about the single die.
4:37So the simple space
4:41will be 1 2 3 4 5 6.
4:49Okay?
4:51Cuz a die has six faces.
4:56You're talking about uh even.
5:00So even
5:03let's look for even number
5:08that's two
5:114 6.
5:14So n even=
5:203.
5:23Got it?
5:29the
5:31and then the second event is prime. So
5:34prime
5:38looking for prime numbers will be two
5:433
5:46five.
5:50So n prime
5:57there are also three outcomes.
6:07Is this clear?
6:12>> Yes.
6:13>> Now look at the two events.
6:17We got even
6:20the outcomes are 2, four and six.
6:26Crime the outcomes are two, three and
6:31five.
6:32What do you observe? Do you think these
6:35two events
6:37have something in common?
6:44>> Yeah.
6:46What is this?
6:48>> They both equal three.
6:51>> Not only look at the outcomes
6:56and they both have two.
6:58>> Yes. So these two events are not
7:03mutually exclusive
7:05because they have number two
7:09and comment. So therefore, we're going
7:11to put even sorry
7:16even
7:21and prime
7:29will be just two.
7:34Make sense?
7:36>> Yes.
7:38and then n
7:43even
7:48n prime. I'm going to use e for event.
7:54I'm going to use e for event and then b
7:57for prime. Okay.
8:02So, e
8:04event
8:08and pi
8:10will be just one because there's only
8:12one I would call.
8:17Got it.
8:20>> Yes.
8:21>> Now,
8:25now you're going to
8:29try
8:39The addition
8:45of probability.
8:57So recall
9:01P of A or B
9:07equals P of A
9:11plus P of B
9:14minus P of A and B.
9:21Now you're going to replace a
9:25[clears throat] with even or e and then
9:29replace b with
9:32prime or p.
9:47P of
9:49even if or even or
9:54or prime
9:56would be P of even
10:00plus P of prime
10:03minus P of even
10:08and prime.
10:12Is this clear?
10:21Yes.
10:26>> Mhm.
10:28>> Yes.
10:29>> Now.
10:30>> Mhm.
10:38Now,
10:41so
10:48n P of A will be E. P of E will be N of
10:52E
10:54over N of S
10:59plus P of prime will be N of P over N of
11:04S
11:06minus
11:07P of A and P will be N of E
11:13N P
11:16over
11:24Now
11:26n of e is three. n of s is 6.
11:31That's 3 / 6. And then n of e and e will
11:35be 1. That's 1 / 6.
11:41soap
11:48of even
11:52or prime
11:56will be 6 - 1 is 5 5 /
12:15Is this clear to everyone?
12:20>> Yes.
12:22>> This is how you
12:24perform
12:26addition rule of
12:29probability. Excuse
12:43me, professor. I want to make a note.
12:45Can you like scroll this paper down?
12:55I'll give you one minute. Let me see if
12:56I can
12:58squeeze this for you.
13:02All right.
13:05>> Yes. Thank you.
13:07Welcome.
13:33I want part A when you done let me know
13:36to move
13:38part B
13:47the simple space we're talking about
13:52a die. So as you know a die has six
13:58uh faces
13:591 2 6 that's the sample space. So the
14:04sample space got 1 2 3 4 5 6.
14:09So the number outcomes in the sample
14:12space that's n of s is six.
14:45Is this uh clear to everyone?
14:50>> Yes. Yes.
14:52>> Okay,
14:54let's move uh
14:58to part B.
15:01B.
15:08B.
15:12B.
15:19B. Here.
15:26B.
15:30Even or less than five. Even let's use
15:36even.
15:37Even it's 2 4
15:416.
15:42So n of even
15:47equals 3. The second event is less than
15:51five.
15:53Less than five.
15:56So that'll be 1
15:592
16:013
16:073 4.
16:10So n of less than five,
16:14we got four outcomes
16:20right.
16:23>> Mhm. Now,
16:25do you think these two events are
16:28mutually exclusive? Do you think they
16:31have something in common?
16:36You got even
16:38>> two, four, six. You got uh less than
16:41five. You got one, two, three, four.
16:43Right.
16:44>> Yes.
16:45>> So, they have two outcomes in common.
16:48>> Mhm.
16:49>> So, you have uh
16:55uh even
16:59and
17:01less than five.
17:04We got two
17:07and then
17:10two and four.
17:13So n
17:15even
17:19and
17:21less than five
17:24cuz two because you got two outcomes.
17:32Is this clear? Now, now let's find
17:37the probability.
17:50P of
17:53even
17:56or
17:58less than five
18:01will be P of even.
18:06plus P of less than five minus P
18:12of even
18:22and
18:25less than five.
18:46Okay.
18:57B of even is N of even
19:03over N of S
19:06plus
19:08P of less than five will be N of less
19:10than 5
19:12/ N of S
19:15minus
19:17N of even
19:21n
19:23less than five over n of s.
19:37N of even got three.
19:41N of S is 6
19:44less N less than 5 is 4 / 6 minus N of
19:50even and less than five is 2 / 6.
19:56So
20:02P of even
20:07or less than five
20:10would be 5 over 6.
20:20You see over here
20:243 + 4 is what?
20:303 + 4 is seven, right?
20:33>> Yes.
20:34>> You see over here
20:377 / 6 will give you a number that is
20:39greater than one. As you know the
20:41probability cannot be greater than one.
20:43That's why here you need to remove
20:45that's why I put a minus sign here.
20:46Okay? Otherwise you will be in big
20:49trouble here. The probability will be
20:50greater than one. So that's why you need
20:53uh to find the entire section of the two
20:56events which is 206.
21:03Is this clear?
21:05>> Yes.
21:07>> Okay. Good.
21:23Now
21:27the last uh part C. Are you done? Can we
21:30move to part C?
21:33>> Yes.
21:34>> Yes.
21:35>> Huh?
21:39>> Yes. I'm done, professor.
21:41>> Okay. Part C is what? Party is two or an
21:45Okay. Two.
21:49Two or add.
21:53Two or add.
22:03Two
22:05or add is just two.
22:08So n of two is just one.
22:14Add
22:17would be 1 135
22:26equals 3.
22:28So these two events
22:35are
22:38mutually
22:42exclusive
22:49events.
22:52So why? Because uh they have no outcome.
22:55So event two we got just uh two add you
22:59got 135. So
23:03uh so that will give you P of
23:06uh 2
23:09or add.
23:12It would be P of two
23:15plus P of add. You do not need the minus
23:17P of two or add because they have no
23:22outcomes in common. So P of R.
23:29Okay.
23:31So that gives you N of two
23:34over N of S
23:37plus N of R
23:41over N of S.
23:50That's 1 / 6.
23:53And that's 2 / 6.
23:56That's 4 over 6.
24:00So
24:02p of two
24:05or add
24:08will be
24:10two
24:14will be
24:15>> 2 over 3
24:18>> would be 2
24:20over 3.
24:30Okay, that's how you find
24:34you use the addition rule of
24:35probability.
25:55Are you done?
26:01>> Hello.
26:01>> Yes,
26:02>> you're done.
26:03>> Yes.
26:04>> Okay, good.
26:06Let's move uh somewhere.
26:13Let's talk about uh
26:18conditional
26:19probability.
26:21So events involving uh
26:26and
26:36earlier to use uh the conditional
26:39probability.
26:42So
26:46some events uh happen
26:50before or will happen. Okay.
27:02Events involving end
27:22If you don't, let me know.
27:39Good.
27:43>> Okay.
27:45Now
27:47conditional see and and conditional we
27:51see the word condition. Okay.
27:54The probability of event B
27:58computed on the assumption that event A
28:02has happened
28:06is called the conditional
28:09probability of B given A. Okay. Any time
28:13they give you a problem you see they ask
28:16for the probability you see given
28:19an event. So you're talking about the
28:22conditional
28:23probability and is denoted P of see this
28:31this is red P of B given A okay P of B
28:40given A
29:20When you're done, let me know and I'm
29:22going to give you the formula
29:26to determine
29:29a conditional probability.
29:43Me too.
29:45>> Huh?
29:46>> I heard
29:48>> I said me too.
29:49>> Not yet.
30:18Thank you.
30:21Okay, this is the formula
30:25conditional quity formula.
30:28The conditional poly of B. See here
30:32given
30:34a
30:36and is given by
30:39probability of b given a. Okay, that's
30:42the way you're supposed to read this.
30:47This is this is red. Probability of B
30:52given A is equal to P of A intersection
30:59or P of A and B over P of A. Okay.
31:05So formula. So here you need to find if
31:08the two events are not mutually
31:12exclusive. That's the entire section of
31:14the two events. All right. So P of A and
31:18B
31:47Are you done?
31:50>> Yes.
31:52>> Okay. I'm going to show you I'm going to
31:55solve a few problems here.
32:04from the simple space S. You can
32:08you can uh copy from the sample space S
32:142 3 4 5 6 7 8 and 9. A single number is
32:19to be selected randomly.
32:23Given the events
32:26A selected number is add
32:31and event B selected number is a
32:34multiple of three.
32:37Find each probability
32:40A P of B. B P of A and B and then C P of
32:49B given A
32:55P of
32:56given A.
33:02So here the simple space has two iter
35:43Are you done?
35:47>> Yes.
35:48>> Okay.
35:50They give you the simple space. 2 3 4 5
35:546 7 8 and nine. So, eight outcomes in
35:57the simple space. Also they give you two
36:01events.
36:02Event A
36:09talking about add. Okay. Let's look for
36:12add numbers in the simple space. That's
36:14three 5
36:177
36:199. So there are four outcomes in event
36:24A. Event B
36:32selected number is a multiple of three.
36:34Can you from the sample spacer what are
36:39uh what are numbers that are multiple of
36:42three?
36:46What are the numbers
36:50are multiple of three? You got three. 2
36:53is a multiple of three because three can
36:55divide by two evenly. 6 is a multiple of
36:57three and 9 is also so n of b
37:03equals three. Got it?
37:06>> Okay.
37:07>> Now we look at the two events. What do
37:09you observe? Do you think they are
37:11mutually exclusive events?
37:15>> Yes or no?
37:18>> Um
37:19>> no. three
37:21>> and nine
37:22>> and nine.
37:25So n of uh a
37:30and b = 2
37:34right?
37:37>> Yes.
37:38>> Is this clear?
37:40>> Mhm.
37:42>> What they ask to find? Find each
37:45probabilities. The first one
37:47they ask you to find
37:51P of B.
37:54P of B
38:00will be N of B
38:03over N of S. [snorts]
38:07N of B is 2 over
38:128. N of S is 8.
38:20If you want, you can simplify here.
38:23That's 1 / 4.
38:27That's your P of B.
38:38So the two events are not mutually
38:40exclusive because they have two outcomes
38:47in common. The two outcomes are three
38:51and 9
38:56>> professor.
38:57>> Yes.
38:59>> I'm confused. Isn't n of b equal to
39:02three?
39:06Yes, you are.
39:10This is just two over eight. Thank you.
39:13Done here.
39:19Let me just remove this. Thank you.
39:26This 3 over 8. I know I have to this.
39:31Okay.
39:36Now they also find P of A and B.
39:49E of A
39:53and B
39:56will be N of A and B
40:02or N of S.
40:05So that's two or 2 over 8
40:10simplifying.
40:14So
40:16p of a
40:19and b
40:21equals uh 1 over 4.
40:24We simplify here.
40:52got it.
41:07>> Is this clear?
41:10Yes.
41:14>> Now the last part here is part C.
41:18Part C
41:28it has to find P of B given A.
41:35So that will give you P of A
41:40and B
41:43over P of A. That's the formula I gave
41:45you.
41:47All right. We don't we do not know the
41:50value of P of A. Let's find out.
41:52Determine
41:57P of A.
42:03P of A
42:06would be N of A
42:10over N of S.
42:14Um
42:16N of A was uh four, right?
42:26Yeah. Four.
42:32That's 4 over 8
42:44of B
42:53given A.
42:58We found P of N B before
43:03it was 2 over 8. Okay.
43:07And then this guy is 48.
43:18Now
43:20if you want you can cancel this
43:24but this one. Sorry.
43:34If you want here you can uh get rid of
43:37the eight. So we got only 2 over 4.
43:42So
43:45p of b
43:47given a simplifying here that will be 1
43:51/ 2.
43:58Got it?
44:05Yes.
44:58Is this uh clear to everybody?
45:04>> Yes.
45:10>> Are you done?
45:19Not yet.
45:31P of A is N of A over N of S. N of A we
45:35found that before was four.
45:38>> N of S is eight.
45:41And then P of A and B we found that
45:44before it was 2 over 8 or 1 over 4 but I
45:47use 2 over 8 okay instead of 1 over 4.
45:51And then we found P of P of A is 4 over
45:548. So that's why here I cancel Z8. See
46:00here I cancel I can do that. All right.
46:02So that 2 over 4 you simplify by two. So
46:07the probability of
46:10of B given A is 1 / 2.
46:20Can you please call it up so I can see
46:23please?
46:34Thank you.
46:36>> Is this good?
46:38>> Yes. Thank you.
46:40>> Very cool.
48:04I'm done.
48:18So this one
50:00You done?
50:04>> Yes.
50:05>> Okay.
50:08Uh given a family with two children. So
50:12a couple
50:14who
50:16decide
50:18to have children but just two. Okay.
50:24So find the probability that both are
50:29boys
50:31given that at least one is a boy.
50:36So first
50:37let's uh
50:41solution here.
50:44So let uh
50:49uh
50:55let A equals both our boys.
51:04No, let's say B instead.
51:07Say B.
51:10Both are
51:13both are boys.
51:18And then A
51:20let A be
51:23at least one is a boy,
51:32one is
51:36a boy.
51:38Okay, I'm going to replace uh B
51:42with both are boys
51:45and A at least one is a boy. Is that
51:48clear?
51:49>> Yes.
51:50>> Now,
51:53as you know, anytime they ask to find
51:56the probability here, you need to find a
51:57sample space. Now can someone tell me
52:02how many outcomes would you expect
52:06in the simple space? How many outcomes?
52:16Two
52:18>> set two.
52:20Determine
52:24the simple space.
52:33Okay,
52:36let's see here.
52:40When you talk about children
52:44can be girl or boy,
52:49right?
52:51>> Mhm.
52:52>> Girl or boy.
53:04Girl, boy.
53:08Girl, boy.
53:10So here this is uh
53:14first
53:17charge. So this one is the second.
53:26Okay.
53:28Now the sample space should look like
53:32this.
53:35So the two people
53:41They try
53:45and then the first one is a girl.
53:49The second one can be a girl. So the two
53:52children can be girl. Girl agreed.
53:57>> Yes.
53:58>> Or the two can be the first one is a
54:02girl, the second one is a boy.
54:07or the first one is a boy, the second
54:11one is a girl or both are boys.
54:17So here
54:23four outcomes in the simple space. Okay,
54:27they try.
54:29The first one is a girl.
54:32They try again. The second one is a
54:34girl. That's my case. I have two girls.
54:36I have no boys.
54:39Or the first one can be a boy. The I'm
54:43sorry. The first one can be a girl. See?
54:46Girl, girl,
54:48girl, boy,
54:50boy, girl, or boy, boy. Got it?
54:58>> Yep.
55:02Simple.
55:05Now
55:22sorry
55:24So
55:34now let's find
55:38B. Both are boys.
55:41So B both are boys. So that would be BB.
55:44Yes or no?
55:47>> Yes.
55:47>> So NB
55:49will be one.
55:53A at least one is a boy. At least one is
55:58a boy. So what does that mean? It can be
56:00B
56:02G
56:04B or BB.
56:12So N of A
56:15would be three.
56:18A and B.
56:21Let's look for
56:23would be just boy boy.
56:27So n of a and b
56:33would be just one.
56:35All right.
56:37So look the two events here. BB that's B
56:44A is B GB. So they have one outcome in
56:48common. Boy, boy, boy. Is this clear?
56:54>> Yes.
56:55>> Now,
56:57so what they ask to find find the
57:00probability that both are both are boys.
57:04So remember we use uh uh B for both are
57:11boys
57:14and we use A at least one is a boy.
57:19So the probability here will be
57:25the probability of B
57:29both are boys given at this one is a boy
57:33would be
57:35P of A
57:40and B
57:42over P of
57:50Is this clear?
57:54>> Yeah.
57:58Now
58:03P of A and B would be N of A
58:10and B over N of S
58:15and then P of A would be N of A over N
58:18of S.
58:26So if you want you can get rid of n of s
58:28here.
58:30So that gives you get only
58:34n of a and b
58:41and get rid of n of s.
58:51Now you have only n of a and b
58:59over n of a.
59:03n of a and b we found that before was
59:06one. n of a is three. That's three.
59:12So
59:18>> wait, why is n of a and b one?
59:26here.
59:32>> You see it?
59:34>> Could you explain it?
59:36>> Okay, sure.
59:40Look at uh
59:46look at this set here and this one here,
59:48right?
59:51You can observe they have BB and common.
59:54Can you see that?
59:55>> Yes.
59:56>> So that's why it's one.
59:58>> Okay. Thank you.
1:00:00>> Welcome.
1:00:02So that's uh so
1:00:08p of
1:00:12both our
1:00:16boys
1:00:20given
1:00:22at least
1:00:25one
1:00:27is a boy
1:00:38would be one/ird.
1:00:42That's the financial.
1:00:54>> It's not bad. It's just a lot to get
1:00:56there.
1:00:57>> Yeah. You know, life doesn't go in
1:00:59straight lines.
1:01:02Life goes on walking lines.
1:01:07Yes or no?
1:01:11H.
1:01:13>> Yeah. Like a roller coaster.
1:01:15>> Yeah. Yeah. Yeah. Okay. Now,
1:01:20if you're done, we're going to come back
1:01:22to to our PowerPoint.
1:01:27Continue here.
1:01:31It's how you find conditional. Anytime
1:01:33you see the word given, so you you are
1:01:38dealing with a conditional
1:01:42probability.
1:01:43Okay? anytime. See that word?
1:01:47All right.
1:01:50Now, if you're done, let me just move
1:01:54here
1:01:56to
1:02:01here.
1:02:07Now, let's talk about independent event.
1:02:18Two events A and B are called
1:02:21independent events if knowledge
1:02:25about the occurrence of one of them has
1:02:29no effect
1:02:31on the probability of the other one.
1:02:34That is if the quality of B given A
1:02:42equals P of B or equivalently
1:02:48the quality of A given B equals P of A.
1:02:54If this
1:02:56happens,
1:02:58the two events A and B are incident.
1:03:13Got it.
1:03:20Okay,
1:03:23you're done. Thank you.
1:04:47You done?
1:04:52>> Yes.
1:05:11Okay,
1:05:13let me show you an example.
1:05:16Just give me one minute.
1:05:23Okay.
1:05:26A single card is to be drawn from a
1:05:29standard
1:05:3152 card deck.
1:05:34Given the events,
1:05:37the selected card is an ace.
1:05:41B. The seed card is red.
1:05:47Find P of B.
1:05:50B. Find P of B given A.
1:05:55C. Determine whether events A and B are
1:05:59independent.
1:06:01All
1:06:03right. So again copy here.
1:06:07So here you can observe
1:06:13that 52 card. So the sample space here
1:06:17is just
1:06:1952.
1:08:13Are you done?
1:08:24No.
1:08:28Are you still there?
1:08:30>> Yeah.
1:08:31>> Are you done?
1:08:33>> Yes.
1:08:35>> Okay.
1:08:37So,
1:08:39we talk about cord before, right? So,
1:08:47the simple space here got 52 card.
1:08:51So, the number outcomes is just 52. They
1:08:54give you two events.
1:09:03Cord is an ace. Okay. So that's a
1:09:08is an ace.
1:09:13Talking about ace.
1:10:01Okay, I got ace of art, ace of diamond.
1:10:05They are red. And we got ace of spade is
1:10:09black and ace of what is the other one
1:10:11again? Off boy.
1:10:18Ace of uh
1:10:22spade.
1:10:24I I forgot the other one.
1:10:30Anyone? You got ace of heart, ace of
1:10:34diamond, ace of spad.
1:10:37Okay. Ace of club.
1:10:40Okay. Let me here.
1:10:43Ace of club.
1:10:46So here you have n of a get four.
1:10:54Got it.
1:11:02Hello. Are you with me?
1:11:05>> Yes.
1:11:10>> The people are quiet today. Only one
1:11:12person.
1:11:15saying yes.
1:11:18Let me call some names here. Ina, are
1:11:21you there?
1:11:25>> Na.
1:11:28Alissa
1:11:31>> here. Professor
1:11:35>> uh
1:11:38>> I'm here.
1:11:44Are you there?
1:11:46>> Yeah, professor. I was drinking water.
1:11:51>> I'm here.
1:11:56>> I'm here.
1:11:56>> Okay, good. Now, you know a 52 card deck
1:12:01has four aces. You got ace of diamond,
1:12:06ace of heart. Both are red.
1:12:11You got ace of spad and ace of club both
1:12:16are black. So therefore the number
1:12:20outcome and uh event a is four. Now
1:12:26about b
1:12:34code is red. So, how many red cards do
1:12:38you have any
1:12:40standard 52 card deck? Anyone? We'll
1:12:42talk about this last time.
1:12:52Okay, talking about red card, there are
1:12:5726 red card. 12 are
1:13:03uh the heart we got 13
1:13:08diamond we got 13 that's 26
1:13:10but what I'm going to do here
1:13:14I don't think I'm going to give you all
1:13:17I'm going to put ace of art
1:13:22and ace
1:13:24of diamond
1:13:27and then of plus
1:13:30we got 26. So n of
1:13:34b
1:13:36= 26.
1:13:3926.
1:13:44Is this clear?
1:13:50There are 26 red cards. 13 uh
1:13:5513 hearts, 13 diamonds. That's 26. But
1:13:59two of them got ace of heart and ace of
1:14:05diamond.
1:14:07We got seven, got eight. We got nine,
1:14:1110. But I I don't have to give you all
1:14:15comes here. I only give you these two
1:14:17because I need these two to show you the
1:14:20two events are not mutually exclusive
1:14:23events. Okay. So here
1:14:32so A
1:14:34and B
1:14:39will be
1:14:45diamond
1:14:47is of art.
1:14:57N of
1:15:00A
1:15:02and B
1:15:04= 2.
1:15:26Is this clear?
1:15:32Hello.
1:15:36>> Yes, refer.
1:15:37>> You're not paying attention. Maybe you
1:15:39are doing something else. I'm going to
1:15:41ask you to open your cameras because
1:15:43you're not going to ask you to open your
1:15:45cameras because we need to do this
1:15:48together. Okay.
1:15:51Is this clear to everyone?
1:15:54>> Yes. Yes.
1:15:55>> Okay. Now,
1:15:57>> what they ask you to find? Find P of B.
1:16:02Find P of B.
1:16:05So to find P of B very simple here. P of
1:16:08B
1:16:12will be N of B
1:16:16over N of S.
1:16:18What is N of B? 26
1:16:21and off first you get 52.
1:16:24So simplifying here P of B is 1 / 2
1:16:29simplified by
1:16:40part B
1:16:44that's to find P of B given Hey,
1:17:12right
1:17:14down.
1:17:17P of A will be N of A and B
1:17:24over N of S. That's P of A and B. P of A
1:17:29would be N of A
1:17:31over N of S.
1:17:34So if you want here, you can get rid of
1:17:36N of S.
1:17:43So you have P
1:17:45of A even P of B
1:17:54given A
1:17:56would be N of A and B
1:18:02over N of A.
1:18:07n of a and b. We found that before it
1:18:10was 2.
1:18:13N of a is uh
1:18:17um
1:18:19is four.
1:18:25So
1:18:30b of b given a simplifying here that
1:18:34would be 1 /2.
1:18:39Now the last thing they ask you in part
1:18:41c
1:18:43they ask you if a and b are
1:18:47independent events. Okay
1:18:51I would say yes. I would say
1:18:55see yes
1:18:59they are
1:19:06independent event.
1:19:15Why?
1:19:19Because
1:19:24the probability of B
1:19:29given A
1:19:31is equal to P of B which is 1 /2.
1:19:43Okay. So they are the two events are
1:19:46independent. Why? Because
1:19:49the probability of B given A equals P of
1:19:54B.
1:20:01Is this clear?
1:20:05>> Yeah.
1:20:06>> Yeah.
1:20:08First stretch
1:20:09>> are independent because the pro of B
1:20:13[clears throat] given A is equal to the
1:20:18B
1:20:37Got it.
1:20:44>> Yes.
1:20:46>> Let let me cover one more thing here in
1:20:49chapter 4.
1:20:53All right.
1:21:00Yes, they are independent events because
1:21:04the probability of B given A is equal to
1:21:07the polarity of.
1:21:12Let's talk about events involving and
1:21:19if you multiply
1:21:21both
1:21:24both sides of the conditional
1:21:27probability
1:21:30by P of A. We obtain an expression for
1:21:35P of
1:21:38A
1:21:40and B.
1:21:44= P of A
1:21:46* P of
1:21:49B given A. So that that's what you're
1:21:52supposed to get.
1:21:57If the two events are independent,
1:22:01you can replace P of B given A with P of
1:22:06B. actually trying. So the calculation
1:22:09of P of A and B is simpler when A and B
1:22:15are independent. So if they are
1:22:18independent you can replace
1:22:24P of A and B
1:22:27that will be P of A
1:22:30times
1:22:31P of B because if the two events are
1:22:35independent P of B given A is equal to P
1:22:39of B.
1:22:44Got it?
1:22:48Yes. Yes.
1:23:44distribution rule of probability. If A
1:23:47and B are any two events, then P of A
1:23:51and B equals P of A * P of B given A. If
1:23:56a and b are independent then p of a and
1:24:01b is equal to p of a * p of a
1:25:50I'm done.
1:25:53>> Yeah. Let me show you two
1:25:56simple examples here and then we take a
1:25:58break.
1:26:00Uh when we
1:26:02come back
1:26:04I will start chapter five.
1:26:07Okay.
1:26:10Yeah.
1:26:14Jeff
1:26:16draws balls from the jaw below. Okay,
1:26:21he draws two balls
1:26:25without replacement.
1:26:28Find the probability that he draws a red
1:26:33ball
1:26:34and then a blue ball in that order. So
1:26:38there are four red balls, three blue
1:26:42balls, two yellow balls. Okay, I would
1:26:44for
1:28:21All
1:28:34right. Good.
1:28:37>> This is very simple here. You don't have
1:28:39to put that down. I will show you the
1:28:41solution in a few minutes. Okay. So, you
1:28:45have uh
1:28:47the to bowl here. You have uh
1:28:53I'm going to show you the solution. Just
1:28:55give you one minute. So here you have uh
1:29:03you have
1:29:05nine balls. Okay.
1:29:09So here they have to find P of
1:29:14uh two balls. The first one is a red
1:29:17ball. So red
1:29:22and
1:29:24a blue that's the second ball. So that
1:29:27would give you P of A
1:29:30sorry P of O as a red ball
1:29:43times
1:29:45peak of
1:29:48uh
1:29:50blue given is a red.
1:29:54Okay.
1:29:56Now
1:29:58because
1:30:00this is a keyword here with I would
1:30:04replacement.
1:30:07Okay. So when Jeff uh
1:30:12draws the ball,
1:30:14he does not put the ball back in the
1:30:17jaw. Is that clear? He does not put the
1:30:20ball back.
1:30:23Are you with me?
1:30:26>> Yes.
1:30:26>> Yes.
1:30:27>> Because of this word, the two events
1:30:31are not independent. Okay.
1:30:36because you put the ball back and not in
1:30:39spend it. So very simple here. What you
1:30:42going to do here? How many red ball do
1:30:43you have?
1:30:48>> Four.
1:30:50>> Four over what?
1:30:54>> Nine.
1:30:56>> This is the red ball, right?
1:30:59Times
1:31:01the second ball is a blue ball. How many
1:31:03blue ball you have?
1:31:06Three
1:31:08>> over what?
1:31:16>> Nine.
1:31:17>> H
1:31:18>> nine. No, eight cuz you didn't put it
1:31:20back.
1:31:21>> Ah, very good. So, here should be over
1:31:25eight because
1:31:28suppose uh
1:31:31it shows uh this uh red ball here. Okay.
1:31:35All right. That's the first ball. So,
1:31:38how many we got left? We got eight. So,
1:31:40it should be three over eight because
1:31:41not put the ball back. Is this clear?
1:31:47Got four red over the total ball. Got
1:31:51nine balls in total. And then also the
1:31:55second ball is a blue ball. This is the
1:31:58first ball. That's the red ball here.
1:32:02The second ball is a blue ball.
1:32:06But
1:32:08once you draw a ball, you cannot put the
1:32:12ball back and the jaw. So here, this is
1:32:16the red ball. You got it left. Okay. So
1:32:20is this clear to everyone?
1:32:22>> Yes.
1:32:23>> Let me show you this.
1:32:25You go.
1:32:33Okay, the probability the first ball is
1:32:36a red ball. That should see R stand for
1:32:39red. That's the first ball. And the
1:32:41second ball is a blue ball. All right.
1:32:46So that's P of R1. That's the first
1:32:48ball. Two events are not independent. So
1:32:51that's uh 49 * 28. So that's uh 1 / 6.
1:33:01got it.
1:33:03>> Yes.
1:33:06>> Got it.
1:33:10>> And so the two events are
1:33:13not independent
1:33:15because they put the ball. Sorry, they
1:33:18did not put the ball back in the jar. If
1:33:23they put the ball back in the two events
1:33:28are independent. Okay.
1:33:42>> You done?
1:33:46>> Yes.
1:33:48>> Okay.
1:33:48>> Yeah.
1:33:49So same problem here but there's a just
1:33:53[clears throat] a
1:33:54uh difference here the difference is
1:33:58this
1:34:01this time with replacement.
1:34:06Okay.
1:34:07With replacement
1:34:13before without replacement the two
1:34:16events are not independent.
1:34:19Now with replacement the two events are
1:34:23independent. Okay. So here that will
1:34:26give you very simple here. Same thing
1:34:28here. So that gives you
1:34:31P of
1:34:33red. That's the first ball. And P
1:34:39would be P of O1
1:34:44times P of B2 because the two events are
1:34:48end dependent. Red ball got four red
1:34:53ball that's four over 9.
1:34:56You got three blue ball but over because
1:35:00you put the ball back. That's
1:35:02difference.
1:35:07So here that would give you let me see
1:35:10here
1:35:114 over 27.
1:35:16So here the two events are independent.
1:35:29Is this clear?
1:35:33>> Yes.
1:35:35>> Let me show you the solution here. It's
1:35:37okay. I'm going to show you the
1:35:38solution. Here you go.
1:35:41Because the board is replaced,
1:35:44repetition are allowed. In this case,
1:35:48event B2 is independent of
1:35:52R1. All right.
1:36:09So, let's take a
1:36:19It's 7:38 now. Let's take uh a break. We
1:36:24will uh resume at 7:50. Okay.
1:36:30>> Okay.
1:36:31>> All right. 7:50.
1:36:34>> Thank you.
1:36:35>> Welcome. So,
1:43:17in order arrangement of objects. Okay,
1:43:22there is order in time you we talk about
1:43:26with permutation
1:43:28order is important. Order does matter.
1:43:33Okay.
1:43:35So the number of different uh
1:43:39permutation of n distant object is n
1:43:43factorial.
1:43:46This symbol
1:43:49when you see this
1:43:54is read n factorial. It's just a number.
1:43:59All right.
1:44:01So n factorial is equals to n
1:44:06times you you decrease n by one by two
1:44:11by three and so on. Okay.
1:44:15For example if you have uh
1:44:19five factorial
1:44:22just a number here. See five factorial.
1:44:25Five factorial is five.
1:44:29You subtract five by one, that's four.
1:44:32Subtract four by 1, that's three * 2 *
1:44:361. And then you multiply them. That's
1:44:39120. It's just a number.
1:44:456 factorial
1:44:47will be 6 * 5 * 4 * 3 * 2 * 1. That's
1:44:54720.
1:44:59So, okay.
1:45:23Is she spirit?
1:45:26>> Yes.
1:46:20You done it.
1:46:23>> Yes.
1:46:25>> Yes.
1:46:38Now
1:46:40when you see zero factori is one
1:46:45one factori is also one
1:46:49zero factori is what?
1:46:53One it's just u something you need to
1:46:58accept.
1:47:01All right. To prove this uh you need to
1:47:04take a course I took when I was in
1:47:07graduate school to prove zero factor is
1:47:10one. All right. But just anytime you see
1:47:13zero factor u replace zero factor u with
1:47:17one. Is that clear?
1:47:24Hello.
1:47:27>> Yeah. See to you is what?
1:47:34Hello.
1:47:37>> We need to work here together. When I I
1:47:40need to hear your voice.
1:47:45Okay. I show you this example before. 6
1:47:48factorial is 6 * 5 * 4 * 3 * 2 * 1.
1:47:52That's 720. 4 factorial is 4 * 3 * 2 * 1
1:47:59is 24.
1:48:06>> Yeah, professor.
1:48:08>> Okay.
1:48:10Now let me show you something here.
1:48:14Let me show you an example here.
1:48:19Suppose you have uh
1:48:256 factorial
1:48:28over
1:48:303 factorial.
1:48:32What is 6 by 3?
1:48:39>> Two.
1:48:40>> Two. What is six factori?
1:48:45>> Two factorial.
1:48:47>> That would be wrong. It's more than
1:48:49that. six factorial over two factorial
1:48:52you okay let me just put this up there
1:48:56>> oh I see I got it I see
1:48:59so
1:49:00>> that equals to two factorial okay
1:49:02>> so what can you do here you can take 6 *
1:49:065 * 4 and then you stop at three
1:49:10factorial
1:49:12>> and then you simplify
1:49:18>> okay why I example. It goes all the way
1:49:20down to two and one.
1:49:25>> Yeah, that's uh
1:49:29>> you see the six factorial six factorial
1:49:32here.
1:49:33>> Yeah.
1:49:33>> See this number here?
1:49:35>> Yeah.
1:49:35>> Let me show you
1:49:41now.
1:49:45When you have six factorial,
1:49:50we can go all the way down to
1:49:54to one.
1:49:57That's why we're trying to to show you
1:49:59to show you here. Okay.
1:50:02>> Mhm.
1:50:03>> Got it.
1:50:04>> Mhm. Now if you have six factorial
1:50:09over 3 factorial you can go all the way
1:50:13down here to 6 * 5 * 4 * 3 * 2 * 1 and
1:50:20then 3 factorial can be 3 * 2 * 1
1:50:25>> and then what can we do here is simplify
1:50:28you see here simplify the three simplify
1:50:31the two
1:50:32>> oh I see okay I get it
1:50:34>> the
1:50:35Okay.
1:50:36>> And then you have only
1:50:406 * 5
1:50:44>> * 4
1:50:45>> * 4. Okay. 6 * 5 * 4. So that give you
1:50:5130. That would be 120.
1:50:54Okay. That's uh 6 factorial
1:50:57over 3 factorial. But you cannot say six
1:51:00factori over three factor is two factor.
1:51:04That's wrong because two factor will be
1:51:062 * 1 which is two. But here you can
1:51:09observe is not two. It's just a big
1:51:10number is 120. Got it?
1:51:12>> Yeah. Gotcha.
1:51:14>> Now another way you can do that. If you
1:51:17have six factorial over three factorial,
1:51:22you don't have to go all the way down to
1:51:24one. You can stop.
1:51:28All right, give one minute.
1:51:40Okay.
1:51:42and go all the way down to
1:51:46I'm gonna stop at three
1:51:51because the bottom number here is what
1:51:54is three factorial
1:51:56and then I cancel the three.
1:51:59See here I still get uh 6 * 5 * 4.
1:52:05Is this clear?
1:52:08>> Yeah. If the bottom number were four
1:52:14factorial,
1:52:16I would stop at four factorial.
1:52:22If the bottom number were five, I would
1:52:26stop at
1:52:28five factorial.
1:52:30Is this clear?
1:52:33>> Yeah.
1:52:34>> Got it.
1:52:37Now let's continue here.
1:52:43So the objective of a 9 * 9 sudoku
1:52:48number also is to fill the grid so that
1:52:53each row each column and each three * 3
1:52:57grid contains the digits one to nine.
1:53:00How many different ways can the first
1:53:05row first row
1:53:09times nine pseudo grid be? This is just
1:53:14a 9 factorial here. Yeah,
1:53:17that's the answer.
1:53:19So the number of permutation is 9
1:53:21factorial. So that would give you
1:53:24362,880
1:53:29ways to for this uh
1:53:35different way and
1:53:38way the first of the blank n
1:53:44knife activate.
1:53:52Okay.
1:54:41You're done.
1:54:44>> Yes.
1:54:45>> Yes.
1:54:47>> Continue here.
1:54:48>> Yeah.
1:54:48>> The formula you need to use to find the
1:54:51number of permutation is this formula. N
1:54:56P R is N factorial over N - R factorial.
1:55:03So keep in mind anytime we talk about
1:55:05the permutation
1:55:07order
1:55:10is important.
1:55:18Okay. Now and always
1:55:23greater than
1:55:26uh greater or equal to
1:55:30all. Okay. N
1:55:34cannot be greater than n but they can be
1:55:37equal but n is always greater than or
1:55:40equal to or. Got it?
1:55:44Okay.
1:56:56done.
1:57:01>> Yes.
1:57:03>> Let me show you some examples how you
1:57:05can calculate the number of permutation.
1:57:09Very simple.
1:57:14Find the number of ways of forming four
1:57:19digit codes. Okay.
1:57:22And which no digit is impeded.
1:57:27So
1:57:29when you talk about fourdigit code here
1:57:33can be one 3 five 7 is a fourdigit
1:57:39codes. Make sense?
1:57:42Because you said the no digit is
1:57:45repeated. It can be uh three 0 4 5. Got
1:57:52it?
1:57:54But you need to know again four. Now how
1:57:57many digits are there?
1:58:00>> How many digits are there? How many?
1:58:05>> Four. H
1:58:10>> four.
1:58:11>> No, you need to you need to choose
1:58:16among the digits
1:58:19four.
1:58:21Okay. So here they give you all.
1:58:27O= 4
1:58:29but n
1:58:31equals 10. Why I said 10? Because there
1:58:34are 10 digits. When you talk about
1:58:36digits, talking about number 0 1 2 3 4 5
1:58:426 7 8 9. Okay, they are digits.
1:58:49Is this clear?
1:58:50>> So, professor, it should be single
1:58:53digit.
1:58:55>> Anytime you talk about digits, the
1:58:57digits are 0 1 2 3 4 5 6 7 8 9. So,
1:59:02there are 10 digit that's why n is 10.
1:59:06Okay. Thank
1:59:07>> you're going to choose among the 10
1:59:11four. So you need to know many. Okay.
1:59:14Very simple here. What you're going to
1:59:15do here
1:59:18you're going to
1:59:24you're going to use permutation because
1:59:28you cannot repeat a digits. Okay.
1:59:32So that you use permutation here.
1:59:34n
1:59:36is n factorial
1:59:39over n - r factorial.
1:59:43What is n? n is 10.
1:59:46What is o? O is 10. R is four.
1:59:52Okay.
1:59:56>> Okay.
1:59:57>> Got it. And then you solve this. So let
1:59:59me just show you the solution here.
2:00:04So you need to select four digits from a
2:00:07group of 10 because you got 10 digits.
2:00:10Anytime talk about digits talking about
2:00:130 through 9. Okay. So n is 10 or is
2:00:18four. So that's 10 over 10 factorial. 10
2:00:22- 4 that's 6 factorial. Watch. They go
2:00:25all the 10 factorial. They start at 10
2:00:28and then they go all the way down to
2:00:31one. Can you see that?
2:00:35>> And
2:00:36>> yes,
2:00:36>> same thing for the six factorial. Start
2:00:39at six. You go all the way down to one
2:00:41and then you simplify 6 5 4 3 2 1 and
2:00:48then you and you just put this there for
2:00:51you. So here this will give you uh 10
2:00:57* 9
2:01:00* 8 * 7.
2:01:02Okay. And then you multiply 10 * 9 * 7.
2:01:07So that's 10 * 9 is 90. 90 * 56 is 5,040
2:01:15ways. So there are 5,000
2:01:20for the way you can uh
2:01:23uh form a fourdigit
2:01:27code. Okay.
2:01:29With no digit is repeated.
2:01:33So that means a digit I'm sorry a a a a
2:01:38fourdigit code. If you have something
2:01:40like this one, three, one, four, that's
2:01:45not good. Why is not good? Why this one
2:01:48is not good?
2:01:52>> Is there two the same number?
2:01:54>> Because we can repeat the number.
2:01:56>> Which number here you repeat? One
2:01:59>> one.
2:01:59>> So therefore, you cannot use cannot
2:02:02repeat. So later on you're going to use
2:02:06a different
2:02:08formula if you want to repeat a digit.
2:02:11Okay, that permutation. Okay,
2:02:14got it.
2:02:17>> Got it. Yes.
2:02:18>> All right.
2:02:3043 waste cars started the 27
2:02:36500. Okay, 43. Okay, so that's n= 43.
2:02:42You see here, this guy here is 43.
2:02:46That's n. This is n.
2:02:52How many ways can the car
2:02:57finish?
2:02:59First,
2:03:00second
2:03:02and third. So, what is the value of all?
2:03:15Hello. I mean, n is 43. Do you agree
2:03:18with me?
2:03:20>> Yes. How about all?
2:03:24>> O will be first,
2:03:26>> first, second, third.
2:03:28>> O is three. Right.
2:03:31>> So here there is order.
2:03:34So therefore going to permutation. Why I
2:03:37said there's order. There is a first.
2:03:40There is a second. There's a third. So
2:03:44order. Got it? Order. First.
2:03:47>> Yes.
2:03:48>> Second. or third. Now if you have if
2:03:52they said the two cars finish first. So
2:03:57here there is no order here. Can you see
2:04:00that?
2:04:02>> Yeah.
2:04:03>> So here there is order. There is a first
2:04:06like u I don't know uh in United States
2:04:11there is a a president right? There is a
2:04:15vice president
2:04:17and there is someone else I don't know.
2:04:21[laughter]
2:04:22All right.
2:04:25The the president is first or the the
2:04:29vice president is
2:04:32second or the secretary of state is
2:04:36third. I don't know or uh the defense uh
2:04:40secretary is fourth. Here there is
2:04:42order. All right.
2:04:47>> Gotcha.
2:04:48>> Good. Now, so n is
2:04:53is um
2:04:56is is 43.
2:04:58So you need to select three cars from a
2:05:01group of 43. Why you use permutation?
2:05:04Because there is order. There's the
2:05:06first, there's second, there's a third.
2:05:07So here we have 43
2:05:10cos three taken at a time. All right. So
2:05:14that's 43
2:05:18factorial over
2:05:21remember the formula I gave you for is
2:05:24uh the formula is this.
2:05:29It's uh nr
2:05:32is n factorial
2:05:35over n minus r factorial.
2:05:39Okay,
2:05:40replace n with 43
2:05:43or with 3, you get 43 - 3 is 40. And
2:05:48then here if you let me add this for
2:05:50you. Start at 43, you go down 42,
2:05:55you go down 41, and then you stop at uh
2:06:0040 factorial because you got a 40
2:06:03factorial below. And then you simplify
2:06:0640. That's why you got 43 * 42 * 41. So
2:06:11when you multiply this
2:06:14three numbers, you get 74,046
2:06:19ways.
2:06:26Got it.
2:06:29>> Yeah.
2:06:53Now
2:06:55let's talk about distinguishable
2:06:57permutation.
2:06:59The number of distinguishable
2:07:00permutation of n objects where n1 are of
2:07:06one type n_sub_2 are of an order type
2:07:11and so on is given by this
2:07:15n factori
2:07:17over n1 factorial u * n2 factorial u *
2:07:21n3 factori
2:07:23if you got more type u *
2:07:28nk factory. Keep in mind n
2:07:33is the sum of n1, n2, n3, and four and
2:07:38so on. Okay.
2:07:47Thank you. All
2:08:15right. How you done
2:08:18Not yet.
2:09:00Thank you, professor. I'm done.
2:09:02>> Let me show you an example.
2:09:04So, a building contractor
2:09:07is planning to de develop a subdivision
2:09:10that consists of six onestory. You can
2:09:14take notes. Four twostory
2:09:18House houses and two state level houses.
2:09:22And how many distinguishable ways can
2:09:26the houses be arranged?
2:10:52When you're done, let me know.
2:11:41Are you done?
2:11:44>> Yes. Yes. Yes.
2:11:46>> So this is very simple here.
2:11:50So
2:11:54this is N1. Okay.
2:11:59This is n2 4
2:12:05and then this is n3.
2:12:10So can someone what is the value of n
2:12:12here?
2:12:19n would be the sum of n1,
2:12:22n2 and n3. That's what is that's what it
2:12:27let me go back to this thing here.
2:12:30See
2:12:32n1 + n_sub_2 + n3 will be n.
2:12:37So
2:12:39now n1 is 6,
2:12:42n2 is 4,
2:12:45n2 is two. So n there are 12
2:12:50aes and the subdivision
2:12:54n = 12 n1 is 6 n2 is 4 6 + 4 is 10 n + 2
2:13:03is 12. So that will give you 12 factori
2:13:07over n1. n1 is 6. 6 factori. N2 is 4
2:13:12factori. And then n3 is two. And then
2:13:16when you do this uh uh calculation here,
2:13:20let me just uh show you can do this
2:13:24here. So what are we going to do here?
2:13:27We're going to take 12 * 11
2:13:31* 10
2:13:33* 9 * 8 * 7 and then stop at 6
2:13:38factorial. Why we stop at stop at six?
2:13:41Because six is greater than four. So 4
2:13:45will be 4 * 3 * 1. 2 would be 2 * 1.
2:13:52Now we can show the six. Okay.
2:13:56Now when you multiply the top you're
2:13:58going to get a number. When you multiply
2:14:00the bottom that will give you 24
2:14:04* 2 that's 48. Multiply the top that
2:14:06will give you a number over 48. So when
2:14:09you do the division, you get 13,860
2:14:14distinguishable ways.
2:14:25Is this clear?
2:14:29>> Yes.
2:14:31Yes.
2:14:33Okay.
2:14:35That's how you use your exist
2:14:37permutation,
2:14:42right? Let me show you another example.
2:14:55Determine the number of distinguishable
2:14:59arrangements of the letters of the word
2:15:03initially. Okay. B me.
2:15:08I wait for you.
2:17:31Are you done?
2:17:34>> Yes.
2:17:35>> Okay.
2:17:35>> Yeah.
2:17:36>> You see the the first one part A.
2:17:40You see the word initially, right?
2:17:44How many letters you have in this word?
2:17:57>> Are you sure it's eight?
2:18:01>> How many letters you have in E? Surely
2:18:05nine.
2:18:05>> Nine. Right.
2:18:09So, so the total number here
2:18:14let's put n= 9. Right? Now look at the
2:18:18word
2:18:20you can observe
2:18:22letter I.
2:18:24How many I see you see there?
2:18:32>> Three.
2:18:33>> Three. So n1 is three. about L
2:18:39>> two
2:18:41>> two
2:18:42I N1 is three because you got three IC's
2:18:48n2 is two because you got two L's.
2:18:52So here that will give you
2:18:55n factorial over n1 factori
2:18:59* n2 factorial
2:19:02that's 9 factorial
2:19:05over 3 factorial
2:19:08* 2 factorial
2:19:11and then you do this uh calculation
2:19:13here. So
2:19:21is this clear?
2:19:25>> Yes.
2:19:26>> Yes.
2:19:27>> Let me show you the solution now cuz you
2:19:29have an idea here. Why n is nine?
2:19:31Because you got nine letters in the word
2:19:34initially. Why n1 is three? Because you
2:19:37got three ices.
2:19:39n2 is two. Why n2 is two? Because you
2:19:42got two ls. Okay.
2:19:46So, let me show you the solution here.
2:19:48Now you have an idea. Now,
2:19:52see
2:19:53nine letters total. So, that's nine
2:19:57factorial.
2:20:00Three ices
2:20:02and two L's. So, that's 9 factorial over
2:20:043 factorial. 2 factorial. So, when you
2:20:07do this calculation here, you can go all
2:20:09the way down to
2:20:11you. You stop at three factorial. Okay,
2:20:15if you want here can do that.
2:20:20Can go here. 9 factorial would be
2:20:249 * 8 * 7 * 6 * 5 * 4 and then we stop
2:20:32at three because 3 is greater than 2.
2:20:34then times two factorial then you can
2:20:37show that
2:20:40and then you multiply the tops
2:20:43that will give you a number I do not
2:20:45know and then you divide
2:20:48that number by 2 because 2 factorial is
2:20:522 * 1 which is two
2:21:02let me see zero 4
2:21:068
2:21:084 0 6 / 2.
2:21:15Okay.
2:21:20>> Okay.
2:21:22>> Now B Mississippi. Okay. Got that word.
2:21:27Mississippi.
2:21:31Mississippi.
2:21:35Mississippi. Let's
2:21:38This is the word, right? Mississippi.
2:21:42One 2 3 4 5 6 7 [snorts] 8 9 10 11. So,
2:21:48Mississippi, this word has 11 letters.
2:21:53Yes or no?
2:21:56>> Yes.
2:21:57>> How many S's you have?
2:22:02Or
2:22:03>> how about I
2:22:06>> or
2:22:07>> or about P?
2:22:10>> No.
2:22:11>> Okay. So, here.
2:22:21So, here.
2:22:25I'm sorry. I should
2:22:27Let me stop this for you. Sorry.
2:22:39Mississippi.
2:22:52I give you one minute. Let me just move
2:22:56something here.
2:23:04All right. Mississippi
2:23:08n = 11.
2:23:12N1
2:23:13= 4
2:23:174's.
2:23:18N2 =
2:23:214.
2:23:244 ice.
2:23:27and then n_sub_2
2:23:29= 2
2:23:322 ps.
2:23:34Okay,
2:23:36so that give you
2:23:3911 factorial u
2:23:42over 4 factorial u
2:23:45* 4 factorial u
2:23:48* 2 fac.
2:23:51So 11 * 10
2:23:54* 9 * 8 * 7 * 6 * 5 * 4.
2:24:01So we're going to cancel one of the four
2:24:03factorial.
2:24:09Cano one of the four factorial here.
2:24:14So you have
2:24:17you have uh 11
2:24:20* 10
2:24:22* 9 * 8 * 7 * 6 * 5
2:24:28over
2:24:304 factorial * 2 factorial
2:24:37you multiplying the tops.
2:24:39So 4 * 3 * 2 * 1 * 2 * 1. So it's just
2:24:47multiplying the tops.
2:24:54So it's uh
2:24:5911 * 10
2:25:02* 72
2:25:07and so on.
2:25:15That's
2:25:18one
2:25:2066 6
2:25:233 2
2:25:33by 48.
2:25:36So then give it 34,650
2:25:4334,650
2:25:44different ways you can arrange the word
2:25:48nec.
2:25:55>> Okay.
2:26:04So you see my my first name
2:26:09Jean
2:26:11is just four facto. All right here my
2:26:15first name
2:26:18is J E A N.
2:26:22How many different ways that will give
2:26:25you just four. Now you can observe no
2:26:28letter repeats. So that would be just
2:26:31four factory.
2:26:33So four factorial will give you 24. So
2:26:36my first name
2:26:39there are 24 different ways I can
2:26:43arrange my first name. Got it.
2:26:49>> Got it.
2:26:51>> Got it.
2:26:51>> It can be let me show you here if you
2:26:54want. June.
2:26:58Okay,
2:27:10J E A N.
2:27:13So that would be 4 factorial.
2:27:174 factorial is 4 * 3 * 2 * 1 that's 24
2:27:21different ways. So it can be my first
2:27:24name can be G A N
2:27:28can be G E
2:27:30can be G now
2:27:34can be
2:27:58I don't think it makes more names than
2:28:00that.
2:28:00>> Okay. It can be
2:28:03can be giant.
2:28:07can be
2:28:11J.
2:28:16NB G N
2:28:21G N
2:28:23E
2:28:25six or it can be E
2:28:30A G N
2:28:33E A E A A N G
2:28:37or it can be
2:28:41E
2:28:43N
2:28:45A Okay.
2:28:48E A N
2:28:50G E
2:28:54or it can be E G
2:28:58A N
2:29:01A G N A
2:29:05or it can be
2:29:07N A E G
2:29:12N A J E
2:29:16N
2:29:19E
2:29:21J A
2:29:23N E A G
2:29:27N
2:29:30J A E
2:29:35N G E A
2:29:41or it can be
2:29:47Stick a now. A E N G
2:29:53A E G N
2:29:57A G
2:30:01E N.
2:30:03A G N E
2:30:07or it can be A
2:30:11N
2:30:12E G
2:30:14or A N J E
2:30:19that's 24 different ways.
2:30:23Okay.
2:30:26>> Okay.
2:30:28Now let me see here. If I take the name
2:30:31of SA
2:30:33see here SA is S A
2:30:40N A.
2:30:42So here that's n= 5 here. So that will
2:30:46be just 5 factorial over
2:30:50over 3 factorial.
2:30:54Okay. So sana
2:30:58we have five letters in this
2:31:03okay five but here I observe I got a 3 *
2:31:071 2 3 over 3 factorial so I believe uh
2:31:13sana
2:31:16can 20 20 different ways that would be
2:31:19five * 4 start at three 5 * 4 is 20
2:31:24different ways Okay. So now,
2:31:27>> yep, makes sense.
2:31:30>> Let me see that really. See here really
2:31:34no letter repeat
2:31:36two
2:31:38or seven.
2:31:41So that give you seven factorial.
2:31:447 factorial. That would be 720 * 7.
2:31:49720 * 7.
2:31:56So now
2:31:59your name
2:32:01can be arranged your 5,040
2:32:04different ways.
2:32:06All right.
2:32:17Let me see the name of LCA.
2:32:22Same thing here. Seven different ways.
2:32:26Sorry, seven factory. That gives you
2:32:305,000 for All right. Good. All right.
2:32:34Um,
2:32:35>> mine's eight.
2:32:37>> Is it eight?
2:32:38>> Yeah.
2:32:41>> Yeah, that's true. So, no repeat. Wow.
2:32:46>> [laughter]
2:32:47>> Hey, [clears throat]
2:32:4840,000 320
2:32:51wage. Okay.
2:32:52>> Oh my god.
2:32:55>> All right.
2:33:02Uh my last name it's P H A N O R. Six
2:33:06different way that [clears throat] we've
2:33:09720 different ways. My last name.
2:33:13No letter repeats there.
2:33:17All right,
2:33:18let's continue here.
2:33:21Let's talk about combination.
2:33:23Combination the formula here.
2:33:27The the only difference here
2:33:31you multiply by all factorial. You take
2:33:34the same permutation formula here. You
2:33:37add this thing here. You multiply by
2:33:39this here. That's the only difference
2:33:41here. See here this guy.
2:33:44All right. But here when we talk about
2:33:46combination order is not important.
2:33:50Okay. That's why it said here a
2:33:52selection of all objects from a group of
2:33:55n objects without
2:33:59regard to order. So here order is not
2:34:01important. Okay. Permutation order is
2:34:04important.
2:34:06It does matter but combinations order is
2:34:10not important. Okay. It does not matter.
2:34:12All right.
2:35:04Suppose you need to calculate
2:35:07C 7 C2. Okay. N is 7 or is 2. So that
2:35:11give you 7 factorial
2:35:14over
2:35:167 - 2 factorial
2:35:20* 2 factorial.
2:35:23Okay,
2:35:26got it. Now
2:35:29let's uh simplify here. So here you got
2:35:32five factorial below here because 7 - 2
2:35:35is 5 over
2:35:397 factorial.
2:35:42Take the seven factorial you go all the
2:35:44way down then you stop at five.
2:35:48Two factor is always two and cancel this
2:35:52thing here. So that's 7 * 6 over 2 42 /
2:35:582 is 21.
2:36:01So 7 C2 is what? 40
2:36:0640 is 21.
2:36:12>> Is this clear?
2:36:15>> Yes.
2:36:27Right. So keep in mind combination
2:36:30order is not important.
2:36:34Other does not matter. So can be
2:36:38in so you can have
2:36:42two two people finish first. Two people
2:36:45finish second. So there is no order
2:36:48here. Okay. Is this clear?
2:36:52>> Yeah.
2:36:54>> Right.
2:36:55>> So let's continue here. here. So, a
2:36:58state
2:37:00department
2:37:02of transportation plans
2:37:05to develop a new section of interstate
2:37:10highway and receives
2:37:1316 beads for the project. Okay. 16
2:37:1716 bead for the project. So that's
2:37:22and
2:37:24the state plans to hire four
2:37:29that's all
2:37:31of the winning companies. How many
2:37:34different combination see here? So here
2:37:37we cannot use permutation here. So here
2:37:42you must use
2:37:45I'll give you one minute please.
2:38:19So here n= 16
2:38:22or = 4. So here that would be
2:38:2716 C4. Okay. So 16 C4.
2:38:32When I use the combination that
2:38:34permutation here, that's what they ask
2:38:36you here. He said how many different
2:38:39combination
2:38:42of four companies can be selected
2:38:46from the 16 million companies.
2:38:50Is this clear?
2:38:56>> Yeah. Yeah. And I just use the formula n
2:39:00is 16
2:39:02or = 4.
2:39:09or = 4.
2:39:12Okay. So you need to select four
2:39:16companies
2:39:18from a group of 16.
2:39:21N= 16 or = 4. Order
2:39:26is not important. Okay.
2:39:34really not important.
2:40:39Are you done?
2:40:44>> Yes.
2:40:47>> Huh?
2:40:50>> Someone said no.
2:40:52>> Yes. Yes.
2:40:54Give you a few minutes.
2:41:43Are you done now?
2:41:47>> Yeah.
2:41:49>> Okay. So, that's uh
2:41:5216 C4, right?
2:41:56N is 16.
2:41:59R is 4. 16 - 4 * 4 factor 3. 16 - 4 is
2:42:0612.
2:42:08Now you go all the way down to 12
2:42:11because 12 is greater than 4. And then
2:42:13you simplify the 12 factorial. The 4
2:42:16factorial is 4 * 3 * 2 * 1. When you
2:42:22multiply these numbers, you're going to
2:42:24get
2:42:261820 different combinations.
2:42:301,820
2:42:32different combination.
2:42:53Is this clear?
2:42:57>> Yeah, this is good. Okay.
2:43:02Try this one.
2:43:26Teddai student advisory board consists
2:43:28of 17 members.
2:43:31Three members serve as the board's
2:43:35chair,
2:43:36secretary and web master.
2:43:40Each member is equally likely to serve
2:43:43any of the position. What is the
2:43:45probability of selecting at random the
2:43:47three members that hold each position?
2:43:52Okay, I don't think I can solve this
2:43:54problem.
2:43:56So
2:46:56Are you done?
2:46:59>> Yes.
2:47:01>> Okay. So, there's only one fair outcome
2:47:04here.
2:47:06So, we got this 17 members. Let's come
2:47:09back here. 17 members three members
2:47:13serve. So that's uh a permutation here.
2:47:18Why this permutation here?
2:47:21Because there is what order
2:47:27chair that's the first guy secretary.
2:47:31That's the second guy. And the third guy
2:47:33is web master. Is this clear here? You
2:47:36use what? Permutation.
2:47:39Is this clear?
2:47:42>> Yeah. Yeah.
2:47:43>> Good.
2:47:45Now,
2:47:48they ask you what is the point of
2:47:50selecting at random the three members
2:47:52that hold each position. Here you go.
2:47:55First you find
2:47:58there's only one f outcome. There are 17
2:48:01members. You select three among 17. So
2:48:03that gives you 4,80
2:48:06where the three position can be filled.
2:48:08Okay. Now what is the probability? The
2:48:11probab let me just read for you.
2:48:36Done. I took a picture.
2:48:38Okay. Now,
2:48:42now you need to select
2:48:45the three members. Okay.
2:48:48The three members and each can be either
2:48:52the web master, either the uh butcher or
2:48:56the security. That's one over 4,000
2:49:0180. So that's approximately 2 * uh uh
2:49:06that would give you 0.2
2:49:09that's the probability here.
2:49:57All right.
2:50:03I think um we just solved we solved this
2:50:07for a minute.
2:50:10We solve this.
2:50:12All right. So
2:50:15suppose you have 11 letters
2:50:18consist consisting of one M, four I's,
2:50:22four S and two Ps. If the letters are
2:50:26randomly arranged in order what is the
2:50:28words that the arrangement spells the
2:50:31word Mississippi. Okay, there's only one
2:50:34variable
2:50:3611 factorial
2:50:38that one four factorial four is before
2:50:43and that would be one. Okay, same thing.
2:50:46All right, so that's it here for chapter
2:50:484.
2:50:50All right,
2:50:54so let me just start chapter five. I
2:50:58don't know. It's up to you. If you want,
2:50:59you can stop here or we can continue
2:51:02with chapter we can start chapter five
2:51:05next week.
2:51:06>> I think a good digestion of this
2:51:08information would behoove our learning
2:51:12abilities.
2:51:16>> Chapter five is about probabilities,
2:51:19[laughter]
2:51:23right?
2:51:24That's why I told you earlier
2:51:27probability is not easy. So uh
2:51:31today fail that test when
2:51:35they took a test.
2:51:37>> All right. But
2:51:40uh that would be on the high. So we have
2:51:43plenty of time
2:51:46of uh this. Okay. Now so I'm going to
2:51:50stop here. Okay.
2:51:52>> Thank you. I'm going to stop here. When
2:51:55I meet meet you next week,
2:51:58I will start chapter five.
2:52:03Okay.
2:52:06Now,
2:52:06>> before I let you go, is there anything
2:52:08you need to ask?

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This page contains the full transcript of Statistics Chapter 4 by " Nombres Bibliques : Mystères et Réflexions", generated from the public captions YouTube serves with the video. The transcript has 9,566 words across 1,992 segments, with the original timestamps preserved so you can click any line to jump to that moment in the embedded player.

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Use the transcript to take notes, quote the speaker, build a study guide, generate a summary with ChatGPT or Claude via the YouTube Summary tool, or export it as a timed subtitle file with YouTube to SRT. You can also re-open it in the transcriber to translate the transcript into 100+ languages.

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