IB · Mathematics Applications & Interpretation (AI)
AISL 4.8 Binomial distribution
Learn AISL 4.8 Binomial distribution in this free IB Mathematics Applications & Interpretation (AI) video lesson for SL 4.8—Binomial distribution.
Learn AISL 4.8 Binomial distribution in this free IB Mathematics Applications & Interpretation (AI) video lesson for SL 4.8—Binomial distribution.
Learn AISL 4.8 Binomial distribution in this free IB Mathematics Applications & Interpretation (AI) video lesson for SL 4.8—Binomial distribution.
The video lesson covers the binomial distribution, which is characterized by having two outcomes and four essential properties: a fixed number of trials, independent trials, two possible outcomes, and a constant probability of success.
An example involving a basketball player taking free throws is used to illustrate these concepts, with the player having a success probability of over trials. Key calculations include finding the probability of scoring a specific number of successful shots, the expected value given by , and the variance calculated using the formula .
The lesson emphasizes the importance of understanding the properties of the binomial distribution and how to apply them in various scenarios.
00:00Alright everybody, so in this
00:03lesson we're going to do
00:04the binomial distribution. Now the
00:06first thing to note that
00:07binomial distribution has to have
00:10two outcomes. That's what the
00:12bi part of binomial means.
00:13I think bicycle binomial two.
00:16So bi is two, two
00:17outcomes. Think of clipping a
00:19coin. It's either heads or
00:21tails. But it also has
00:23to have these four properties.
00:25We need a fixed number
00:28Now the files have to
00:28be independent. We need two
00:30outcomes and the probability of
00:32success stays the same. Now
00:34the flipping a coin example
00:35works perfectly well here. Well,
00:38as long as you said,
00:39I'm going to flip a
00:40coin 10 times. Now I
00:42have a fixed number of
00:43trials. So the trial is
00:44the flipping the coin. The
00:46trials are independent because it
00:47doesn't matter what your first
00:49flip, what you get for
00:50your first flip, that doesn't
00:51affect what you get for
00:53your second flip, if that
00:54makes sense.
00:56two outcomes, it's either heads
00:57or tails and the probability
00:59of success stays the same,
01:01yes it does, it's always
01:0250 -50 for each toss
01:04of the coin. Okay, now
01:07I actually don't want to
01:08do the tossing coin example
01:10because it's one a little
01:12bit boring and two, I
01:14don't want a situation where
01:15the probability is 50 -50,
01:17I'd prefer a situation that
01:18probably isn't 50 -50, so
01:19I'm going down the free
01:23throw root, so Thomas
01:24going to take 10 free
01:25throws. So that's your fixed
01:28number of trials is 10.
01:29The probability scores I give
01:31in shot is 0 .65.
01:32So we're going to assume
01:34that that probability stays the
01:35same. It's always 0 .65.
01:37So that's like, let's say,
01:41well, a pro MBA player
01:43might have, well, the like
01:45the best free throw percentages
01:46out there, like 0 .9,
01:48or like 90%. So 0
01:50.65 is that any good?
01:52Well, it's okay. It's probably
01:56better than me, to be
01:58honest, but it's probably not
02:00very good if you were
02:00a pro basketball player. Now,
02:02it then says, let X
02:04be the number of free
02:05throws to scored. So we've
02:07a fixed number of trials,
02:08they're 10. The trials are
02:10independent, so we're going to
02:11assume they're independent. You could
02:13argue they're not independent because
02:15if you start scoring, then
02:17your confidence gets higher and
02:18then you're more likely to
02:20score.
02:20or if you miss 5
02:22in a row, your confidence
02:23is destroyed and you're more
02:25likely to miss, that's debatable,
02:27but we're going to assume
02:28they're totally independent. Two outcomes,
02:31yes, you see the score,
02:31we're going to score, are
02:32you isn't going to score?
02:34And then, yeah, we're saying
02:35the probability stays the same.
02:39Okay. Let's, okay, before I
02:41do this, these A to
02:43H questions, I'm going to
02:45show you what, what we're
02:47going to write. We're going
02:47to say X.
02:48is binomially distributed, right? So
02:53this means x, x is
02:55our number of throws. This
02:56is our random variable. This
02:59means it is binomially distributed.
03:01Now n and p, you'll
03:03see here, n is number
03:06of trials, number of trials.
03:11And the p is the
03:14probability probability
03:16the ability of success. So
03:19this is straight from the
03:21formula booklet. You get this.
03:24So, NP, the, how many,
03:26what's N here? How many
03:28trials are we? There's Tom
03:31doing, how many free throws?
03:32Well, it's 10. And once
03:33the probability of success, the
03:34probability scores is 0 .65.
03:39So, just writing that down
03:42will often get you a
03:43mark because it should
03:44you understand the disbinover distribution
03:46and you understand the number
03:48of trials in the probability.
03:50Okay, so let's do each
03:52of these. Luckily, this can
03:55all be done with your
03:56calculator and it's fairly straightforward
03:58once you know where the
03:59buttons are. So what's the
04:02probability that x equals five?
04:05So what this means is,
04:07what's the probability that he
04:08gets five out of ten
04:10free throws? Now you might
04:11think, well,
04:125 out of 10, that
04:16must be 50, 50. Well,
04:20that's definitely not right because
04:22the probability scores are given
04:23the shortest point 6, 5.
04:25So it's not going to
04:25be 50, 50, 50. Or
04:27you might even say, OK,
04:29well, then it's point 65,
04:30but it's not. It's actually
04:33a really complicated problem. And
04:35without a calculator, let me
04:39just let me give you
04:40an idea of
04:40how hard this problem is.
04:42So let's say one mean
04:43T score is zero mean
04:44T doesn't score. What's the
04:46probability gets five? Well, he
04:47could score one, two, three,
04:50four, five, and then miss
04:51five, two, three, four, five.
04:54And we could find the
04:55probability of this happening. It's
04:560 .65 times 0 .65
04:57times 0 .65 times 0
04:59.65 times 0 .35 times
05:010 .35 times 0 .35
05:03times 0 .35 times 0
05:04.35. That would give me
05:05the probability of this happening.
05:07But that's not the only
05:08thing that
05:08happen. I could also, well
05:10Tom could also do this,
05:13miss this one, score that
05:15one, and then miss those.
05:18That's five. But he could
05:19also miss, miss, miss, score,
05:22score, miss, score, miss, score.
05:26Two, three, four, five, six,
05:27seven, eight, nine, ten. Or,
05:30well, how many different outcomes
05:33are there here? There's loads,
05:35there's actually, there's actually
05:36actually 10 choose five outcomes.
05:40Now I'm not going to
05:41get into this because I
05:42want to show you how
05:43easy it is to do
05:43it through calculator. But there's
05:45just many, many, many different
05:47situations that can happen and
05:48you have to find the
05:49probability of all of them.
05:50So it's quite a complicated
05:52problem. But we're going to
05:54make it and I'll show
05:54you how this is quite
05:55easy. So calculator, I'm going
06:00to go menu probability,
06:04and I'm going to go
06:07to binomial, now there's two,
06:10there's two that we're going
06:11to use. binomial PDF or
06:14binomial CDF. PDF stands for
06:17probability distribution function. That's for
06:20when it e like probability
06:21of x equals five. CDF
06:24is for the ones like,
06:26well, part B they're greater
06:27than seven. That will be
06:28CDF are less than three
06:30or between two and six.
06:32That's CDF
06:33because it stands for cumulative
06:34distribution of owners, where you
06:36accumulate a number of outcomes
06:41together. So PDF is when
06:43it equals, and CDF is
06:44when it's greater or less
06:45than. So in this case,
06:47I'm going to do PDF.
06:50The number of trials is
06:5110. That's n. The probability
06:54of success is 0 .65.
06:56We have that. And the
06:57x value is 5. That's
06:59what x equals 5.
07:01Press enter and I get
07:02the answer. It's 0 .15357.
07:060 .1557. Okay, so it's
07:14not likely that he will
07:17score, it's not likely that
07:20he'll score five baskets. But
07:22that's because he might score
07:23four or might score six
07:24or might score three, whatever.
07:26That's the probability that he
07:26scored five exactly. Part B.
07:29The probability that is greater
07:31than 7. Greater than 7.
07:36I'm going to do many
07:37probability. Now you should know
07:39it's also in statistics. If
07:43you press statistics distributions, you
07:45get the same thing. It's
07:46in both. I'm going to
07:48go to binomial with this
07:50time CDF because it's greater
07:51than 7. Now, this is
07:53quite nice because I can
07:55put number of trousers 10.
07:58Probably the success is your
08:00point six five. That stays
08:01the same. Now he gives
08:02me a lower bound and
08:04an upper bound. Now be
08:05careful. If he says greater
08:07than seven, there's a big
08:09difference here between greater than
08:11seven or greater or equal
08:13to seven. If Tom's throwing
08:16some free throws and he
08:17says, I bet you I'm
08:18going to score more than
08:19seven. What's the minimum he
08:22has to score to score
08:23more than seven? Well, that's
08:24eight.
08:25More than seven is eight.
08:30And then the upper bound
08:31is 10, because he's only
08:34taking 10 free throws. So
08:35we're basically finding the probability
08:37scores eight, or nine, or
08:3910 out of 10. Press
08:40enter. There I have it.
08:43Zero point. Two six one,
08:46six oh seven, zero point
08:47two six one six one
08:49six oh seven. That's the
08:51probability that he scores more
08:53than seven.
08:53which would be eight, nine,
08:54or ten free throws. So
08:57I'm doing it quite a
08:59lot of them because there's
09:01quite a lot of different
09:02types that can give you
09:03and I just want to
09:04make sure we do all
09:05of them. So less than
09:06or equal to three. If
09:09Tom said the same thing,
09:11I'm going to do a
09:12menu, probability, distributions. That's quite
09:18annoying. I just clicked the
09:20wrong thing.
09:21probability distributions. Sometimes I actually
09:25like to press the, you
09:28can actually press the number
09:30so 5 and then CDF
09:34is B. Okay, so it's
09:3710, it's 0 .65. Now
09:42if he says, well if
09:45I said Tom I bet
09:46you, I bet you you
09:48score
09:49three, you score less than
09:50or equal to three throws.
09:52Well that means I'm saying
09:54I bet you he'll score
09:54zero, one, two, or three.
09:57So these lower bounds and
09:58upper bounds are important. The
09:59lower bound in this case
10:00is zero. The upper bound
10:01is three. If it's at
10:03less than three, the upper
10:04bound will be two. So
10:06again, be very careful there.
10:07Presentor zero point zero, two,
10:11six, zero, two, four. So
10:17look, that's
10:17like less than 3 %
10:18so it's highly unlikely he'll
10:20score less than or equal
10:22to 3 throws. Which makes
10:24sense because he's got quite
10:25a good shooting average. He
10:27should be scoring more than
10:293 out of 10. Okay.
10:33Part D. What's the probability
10:36that X is greater or
10:38equal to 4? So note
10:39the difference between these two.
10:43TINSPIRE. Menu.
10:45probability distributions and B. Now
10:52I have 10 .65 more
10:57than or equal to 4.
10:59What's the lower bound? Well,
11:014. Because he says more
11:03than or equal to 4.
11:04So 4 is the lower
11:05bound. The upper bound is
11:0610 because he's only taking
11:0710 throws. Okay, 0 .9739760.
11:130 .973, 976. And if
11:19you actually look at this,
11:22less than or equal to
11:223 and more than or
11:24equal to 4, this plus
11:26this actually equals 1. Look
11:29at this, this is, yeah,
11:31this plus this equals 1.
11:33And the reason is because
11:34less than or equal to
11:353 or more than equal
11:36to 4 is everything that
11:38can happen. It's 0, 1,
11:392, and 3 or 4,
11:415, 6, 0, and 10.
11:41Part E. The probability that
11:46x is less than 1.
11:50Okay, what's the probability that
11:52x is less than 1?
11:54Now, yes, you could use
11:57CDF here and your lower
11:58bound is 0, but your
11:59upper bound would also be
12:000. So, the probability that
12:03he scores less than 1
12:04free throw is actually the
12:06same as the probability he
12:07scores 0
12:09So I'm going to find
12:11the probability he scores zero.
12:13So that's 5, 5, b.
12:18Okay, um, right, I can
12:19do it this way. It's
12:20the same thing. Number of
12:21trials is 10. Probability of
12:24success is 0 .65. Now
12:26the lower bound is zero
12:27and the upper bound is
12:28zero. So this is exactly
12:29like doing PDF with x
12:31equals zero. Now I'm expecting
12:32a very low number here
12:35because it's highly unlikely he's
12:36going to miss all the
12:37free throws.
12:38And yeah, look, tiny number,
12:400 .00028, 0 .00028. So
12:47it's highly, highly unlikely, he'll
12:50miss them all. Okay, last,
12:52one of these type X
12:55greater than two, and less
12:58than or equal to six.
13:00Okay, it's CDF, so I'm
13:03going to do menu five,
13:065b so probability cdf because
13:09it's a range of values
13:1010 .65 this is going
13:14to stay the same now
13:15greater than 2 and less
13:16than or equal to 6
13:17these ones are just simply
13:19tricky he has to score
13:21more than 2 so what's
13:22the lower bound 3 and
13:25it's less than or equal
13:26to 6 so the upper
13:28bound is 6 now try
13:31and get that understood in
13:33your head
13:34because that always causes problems
13:36greater than 2 means the
13:38lower bound is 3 and
13:40less than equal to 6
13:40means the upper bound is
13:416. First enter point 481352.
13:46So it's nearly 50 %
13:52that he's going to score
13:53between, well, greater than 2
13:55and less than 6. Okay,
13:56fine. Next bit is the
14:01expected
14:02value. So how much would
14:04do we expect him to
14:06get? Now the expected value
14:08is simply n times p.
14:11Now that's quite easy to
14:13understand why that's the case
14:15or to conceptualize. Think if
14:17I'm tossing a coin and
14:20I flip it 100 times,
14:23how many heads would you
14:24expect to get? And how
14:25many tails would you expect
14:26to get? You'd expect to
14:27get 50 heads and 50
14:28tails. If I
14:30flipped it 500 times how
14:34many heads would I expect
14:35to get? Well 250 heads,
14:37250 tails, which is 500
14:39times P which is a
14:41half 500 times a half,
14:43good, 250. So all I
14:45need to do here is
14:46multiply 10 times 0 .65.
14:52My expected value is 6
14:54.5. I don't round it
14:57to 6 or
14:58seven or anything like that.
15:00The expected value is 6
15:01.5, even though obviously you
15:03can't score, you can't score
15:066 .5 shots out of
15:0810. You either get 6
15:09or 7, but the expected
15:10value is like the mean,
15:11even look, it is the
15:12mean, it says it here.
15:13And remember, if you're getting
15:15the mean of a bunch
15:17of numbers, it doesn't have
15:19to be like the mean
15:21number of siblings in the
15:25class.
15:26you can get 2 .478,
15:30even though obviously you can't
15:31have 0 .478 children. Okay,
15:34and H, this one, the
15:38variance, I'm actually not going
15:41to explain and prove this
15:44to you because it's actually
15:45quite complicated. So I'm just
15:46gonna tell you there's a
15:48formula, there it is. This
15:49is the formula, it's in
15:50the formula of a booklet.
15:51It's the variance, but only
15:52for binomial distribution.
15:54And this is the mean
15:55only for binomial distribution. So
15:58I do one minus p.
16:00And then n is 10.
16:03p is 0 .65. And
16:08I do one minus 0
16:09.65. And I'm just going
16:12to do all that on
16:14my calculator. Let's move that
16:18up here. It's 10 times
16:220.
16:220 .65 times 1 minus
16:270 .65. And I get
16:312 .275. So the variance
16:36is 2 .275. So that's
16:39like a measure of how
16:40spread out his data is.
16:43OK, hopefully that makes sense.
16:46Remember these four properties that
16:49we need in order to
16:50use
16:50the binomial distribution, but they
16:52won't often, they won't tell
16:53you this is binomial distribution.
16:55And in order to realize
16:56it is binomial distribution, think
16:59about the two, that's the
17:00kind of key way of
17:03realizing that you need to
17:05use binomial distribution. It's some
17:07kind of experiment and there's
17:08two outcomes, either he scores
17:10or he doesn't score, either
17:11he wins or he loses,
17:13either it's green or it's
17:15blue, two outcomes. Okay, see
17:18you in the next lesson.
17:18you