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Finding The Probability of a Binomial Distribution Plus Mean & Standard Deviation — Transcript

by The Organic Chemistry Tutor · 2,360 words · 305 segments · language en · Watch on YouTube

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0:00in this video we're gonna talk about how
0:03to calculate the probability of a
0:05binomial distribution so here we have a
0:09problem where a six-sided die is rolled
0:1212 times what is the probability of
0:16getting a four five times so here's the
0:21formula that we need the probability of
0:23getting X so sexist is equal to the
0:28combination formula NC X which can be
0:31written that way times P raised to the x
0:34times Q raised to the N minus X so as
0:40you said before
0:40capital P of X is the probability that
0:43there will be exactly X successes in n
0:47trials lower case P is the probability
0:52of getting a successful event lower case
0:55Q is the probability that an event fails
1:00so in this problem we need to determine
1:04four things n X lower case P and lower
1:09case Q so what are the values of these
1:12four variables
1:14what's n and is the number of trials the
1:19six-sided die is rolled 12 times so n is
1:2312 now how many successful events are we
1:30getting in this problem we want to get a
1:33four five times so we want to
1:36successfully get a four five times so
1:40there's five successful events that
1:42we're trying to achieve here now what is
1:45the probability of getting a successful
1:47event what is the probability of rolling
1:50I 4 using a six-sided die so out of the
1:55six numbers that we can get there's only
1:58one successful number which is the four
1:59so the probability of rolling a four is
2:02one out of six because it's one number
2:05out of the six numbers that we can get
2:08now let's calculate Q
2:11what is it the probability that the
2:12event will fail what is the probability
2:15that we will not get a four out of the
2:19six numbers that we can get five of
2:21those numbers does not represent a four
2:23so the probability of not getting the
2:25four is going to be five out of six now
2:29let's talk about how to calculate this
2:32portion of the equation so we need to
2:35use the combination formula perhaps
2:37you've seen it this is an C R and that's
2:42going to equal n factorial divided by
2:45and the minus R factorial times R
2:49factorial so n is 12 X is 5 so this is
2:5512 C 5 so that's going to equal 12
3:04factorial divided by 12 minus 5
3:08factorial and then times R factorial or
3:13five factorial 12 factorial that's going
3:20to be 12 times 11 times 10 times 9 all
3:25the way to 1 times 8 times 7 factorial 7
3:31factor is basically 7 times 6 times 5
3:33times 4 times 3 times 2 times 1
3:36now 12 minus 5 is 7 so we can write that
3:42as 7 factorial and note that we can
3:44cancel these two 5 factorial that's
3:47gonna be 5 times 4 times 3 times 2 times
3:521
3:53now 4 & 3 they multiply to 12 so we can
3:57cancel and that with the 12 on top 5
4:01times 2 is 10 so what we have left over
4:04is 11 times 9 times 8 11 times 9 is
4:11$99.99 times eight is 792 so this
4:17portion of the formula is equal to 7 and
4:2292
4:27so we have P of X where X is 5 that's
4:31equal to 792 x lowercase P which is 1
4:35over 6 raised to the X power or to the
4:39fifth power times Q which is 5 over 6
4:43raised to the N minus X or 12 minus 5 so
4:52at this point you can just plug this in
4:55to your calculator and get the answer so
5:02just type it exactly the way you see it
5:07and so you should get point 0 2 8 4 2 5
5:15if we multiply that by a hundred percent
5:20we're going to get approximately two
5:22point eight four percent so that's the
5:25probability of getting a four five times
5:29if we roll a 6-sided die 12 times number
5:34two a multiple-choice test contains 20
5:38questions with answer choices a B C and
5:41D only one answer choice to each
5:43question represents a correct answer
5:46find the probability that a student will
5:50answer exactly 6 questions correct if he
5:54makes a random guesses on all 20
5:56questions so go ahead and take a minute
5:59and try this problem so let's begin with
6:03the formula so it's going to be NC x
6:09times the probability that it will be
6:13successful raised to the x power times
6:16the probability that it's going to fail
6:17raised to the N minus X now what are the
6:23variables n X P and Q and it's problem
6:30select if
6:31n is the number of trials in this case
6:32the 20 questions now how many of these
6:3820 trials does a student have to get a
6:42successful event he needs to answer
6:46exactly 6 questions correct out of the
6:4820 questions so there has to be 6
6:52successes in 20 trials now what are the
6:58values of P and Q in this case P is
7:01going to be the probability of getting a
7:03question right by randomly guess in an
7:06answer so there's four answer choices
7:09one of them is correct so the
7:11probability is going to be one out of
7:12four or 0.25 which means the probability
7:17of guessing the question wrong it's
7:20going to be 3 out of 4 because there's
7:23only one correct answer the other three
7:25are wrong three out of four is basically
7:270.75 that's the probability that he will
7:31get the question wrong if he guess
7:32randomly so now that we have the values
7:36of an X P and Q let's go ahead and
7:40calculate the value of NC X or 20 C 6
7:45this is going to be 20 factorial divided
7:48by 20 minus 6 factorial times 6
7:52factorial so that's 20 factorial over 20
7:58minus 6 is 14 and then times 6 factorial
8:03so 20 factorial is going to be 20 times
8:0719 times 18 and we're gonna keep going
8:10until we get to 14 I'm going to leave it
8:14as 14 factorial and then I'm going to
8:20expand
8:20six factorial so we're gonna do just
8:26like we did before in the previous
8:27problem we're gonna cancel 14 factorial
8:30and let's see what else we can do so 5
8:34times 4 is 20 and 6 times 3 that's gonna
8:39be 18 and then 16 divided by 2 is 8 so
8:44what we have now is 19 times 17 times 8
8:49times 15 so if we multiply those four
8:58numbers we're gonna get 30 8760 now
9:06let's calculate the probability of
9:10guessing six answers or six questions
9:13correct out of the twenty questions
9:15so it's 26 which we got it to be thirty
9:19eight thousand seven sixty times P P is
9:230.25 X is six times Q which is 0.75
9:28raised to the N minus X that's 20 minus
9:32six which is 14 so go ahead and plug in
9:36go ahead and plug those numbers in I
9:47made a mistake
9:49I gotta retype everything the answer I
9:58got is 0.16 eight six zero nine I'm
10:04multiplying that by a hundred it's
10:05approximately 16 point eight six percent
10:09so that's the probability of guessing
10:11six questions correct out of the 20
10:15multiple choice question test assume in
10:18this four answer choices if there were
10:19five answer choices P and Q will be
10:22different if there were five answer
10:24choices P would be point 20 Q would be
10:26point eighty it'll be one out of five
10:29four P
10:31number three 25% of all students
10:35enrolled in high school XYZ are taken
10:39algebra 30 students are chosen at random
10:43part a find a probability that exactly 7
10:48students out of the 30 chosen are taken
10:51algebra go ahead and work on this
10:55problem so we're going to use the same
11:00formula as we've been using in the last
11:03two problems so what is n in this
11:12problem n it's going to be the number of
11:16students that are chosen at random
11:18that's 30 X is going to be the number of
11:23successes in n trials so these are going
11:25to be the number of students who weren't
11:28taking algebra out of the 30 chosen so X
11:32is going to be 7 P is the probability of
11:38selecting a student who is taking
11:39algebra and that's going to be 25% or
11:420.25 the probability of selecting a
11:45student who has not taken algebra that's
11:48gonna be Q which is 1 minus P which is 1
11:51minus 0.25 and that's 0.75 so now we
11:58have everything that we need to answer
11:59Part A so we need to count
12:05P of seven so this is gonna be 30 and
12:09seven times 0.25 raised to the seventh
12:15power times 0.75 raised to the 30-7
12:21power so at this point you know how to
12:27use the combination formula to calculate
12:2930 see seven so I'm just going to use my
12:33calculator to get this answer 30 see
12:38seven that is a huge number
12:40it's two million thirty five thousand
12:43eight hundred and we're going to
12:46multiply that by point 25 rate of seven
12:49times 0.75 raised to the 23rd power so
13:02the answer is going to be point one is 6
13:046 to 36 if we multiply that by a hundred
13:10percent the answer is approximately
13:13sixteen point six percent so that is the
13:17probability of selecting exactly seven
13:20students out of 30 who are taking
13:23algebra and who are enrolled in this
13:26score now go ahead and try Part B feel
13:32free to pause the video and work on that
13:35example what is the probability that
13:41fewer than five students out of the 30
13:44who are selected are taking algebra so
13:49now we're not dealing with just one
13:52number but a range of values for X so
13:57has to be less than five which means it
14:03can be one it can be two three or four
14:07but not five it's not less than or equal
14:10to five it's just less than five so what
14:14we're gonna have to do is calculate the
14:16probability when X is 1 2 3 and 4 and
14:19then add them up in order to get this
14:21answer so let's find the probability
14:25that exactly one student out of the 30
14:28is chosen at random who is taken out of
14:31her so and it's gonna be 30 X is 1 P is
14:36so the same it's 0.25 so it's 0.25 race
14:41to the X X is 1 Q is the same 0.75
14:45raised to the N minus X or 30 minus 1
14:49which is 29 so this is going to be 30 C
14:551 hopefully you have a calculator that
14:57can do that times 0.25 times 0.75 raised
15:03to the 29th power and so you should get
15:06point zero zero one seven eight five
15:10eight I multiply that by a hundred this
15:14is approximately 0.1 is seven eighty six
15:19percent now let's calculate the
15:23probability that out of the thirty
15:25students who are chosen exactly two are
15:27taken algebra so this is going to be
15:31thirty C two times point 25 raised to
15:36the second power times 0.75 raised to 30
15:40minus two or to the 28th power
15:53so the answer I got for this is point
15:56zero zero eight six three one five which
16:02is approximately point eight six three
16:08one percent well this should be four six
16:12so I'm going to run it to 0.8 6 3 1 now
16:20let's calculate P of three using the
16:24same process 30 minus 3 is 27 so it's 30
16:35see 3 times 0.25 raised to the third
16:38power times 0.75 raised to 27 power and
16:44this is going to be point zero two six
16:46eight five which is two point six eight
16:53five percent now it was calculate P of
16:57four so it's 30 see four times point 25
17:05to the fourth power times 0.75 to the
17:1026th power so this comes out to point
17:25zero six zero four two so that's going
17:31to be six point zero four two percent so
17:35now what we're going to do is add these
17:40four values to get our final answer
17:54so the probability of getting fewer than
18:00five students who are taking algebra out
18:03of the 30 chosen that's gonna be
18:05approximately nine point seven six nine
18:10percent so if you have an inequality
18:14such as fewer than five you need to
18:17calculate multiple values in this case
18:20so you have to calculate the probability
18:23of getting one two three or four
18:25students out of the thirty who had taken
18:27algebra as you can see this could be a
18:30long repetitive process and if n is very
18:35large you could use the normal
18:38distribution to approximate a binomial
18:40distribution but for this particular
18:43example since we only have to calculate
18:45for I just decided to add all four
18:47values now let's move on to Part C the
18:54final part in this video so how can we
18:57calculate the mean and their standard
19:00deviation of this binomial distribution
19:02how can we do that the mean is going to
19:08equal the number of trials times the
19:11probability of getting a successful
19:14event and in this case is 30 we know
19:19that P is 0.25 so it's 30 times 0.25
19:24which is 7.5 so as you can see it's very
19:31easy to calculate the mean if you know N
19:34and P which is given to you in the
19:37problem now to calculate the standard
19:39deviation it's not that difficult either
19:42it's the square root of n times P times
19:45Q n is 30 P is 0.25 Q is 0.75
19:58now let's go ahead and plug this in so
20:06the answer I got for the standard
20:07deviation is two point three seven one
20:10seven so that's how you can calculate
20:13the mean and the standard deviation of a
20:17binomial distribution so that's it for
20:19this video don't forget to subscribe to
20:21this channel if you haven't done so
20:23already and thanks again for watching

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