00:01
All right, so for this problem, in part a, or actually, just generally, i'll say some things to start off here.
00:08
We know that we're told that the graduation rate is 91 .5%.
00:13
So we can consider the probability of a success here as being 0 .915.
00:18
Looking at all of the different problems, we can also see that they're discussing randomly selecting 11 students.
00:24
So we're dealing with a sample size of 11 for all the different parts of the problem.
00:28
Now, we would have that for each one of the randomly selected students, either they graduate or they don't.
00:35
So we have binary outcomes.
00:39
What this tells us, then, is that if we have x is the number of people in the sample who end up graduating, we would be able to model x as a binomial random variable, where n is equal to 11, and p is equal to 0 .915.
00:58
So what i'll do here is i'll note, first of all, that the probability then that x is equal to any particular value k would be found by taking n choose k.
01:11
So that would be 11 factorial divided by k factorial times 11 minus k factorial times 0 .915 probability of success to the power of k, the number of successes, times one minus or one minus 0 .915, which would give us 0 .085, to the power of the number of failures, which would be 11 minus k.
01:41
So what i'm going to do here for making the calculations for the rest of this problem kind of easy to deal with, i'm just going to bring up my software here, and i'll show you exactly what i'm doing.
01:53
So i'm just going to set up a little function in my software so that i can plug in a particular value of k and get out what the probability would be.
02:01
So this is going to be exactly equivalent to, if you were plugging the values into a calculator manually, i'm just doing it this way to speed up the process a little bit, especially since i want to respect your time here.
02:15
So for part a, we're looking for the probability that at least 11 graduated.
02:21
So that's the probability that x is, or pardon me, at least 10 graduated.
02:25
So it's probability of x greater than or equal to 10, which would be probability that x equals 10 plus probability of x equals 11.
02:36
So we'd have p of x equals 10, so p of 10 plus p of 11, comes out to a result of 0 .761 roughly.
02:51
Then for part b, finding the probability that exactly 8 graduated, i'll fill out the sort of formula here, since this is not too, extensive.
03:02
So we'd have 11 factorial divided by 8 factorial times 11 minus 8, so that would be times 3 factorial, times 0 .915 to the power of 8 times 0 .085 to the power of 3.
03:24
So i'll also note that 11 factorial over 8 factorial times 3 factorial can be rewritten as 11 times 10, times 9 times 8 factorial so we'd be able to divide out the 8 factorial in numerator and denominator divided by 3 times 2 so really that's divided by 6 so we can evaluate that out and that's 11 times 10 times 9 times 0 .915 to the power of 8 times 0 .085 to the power of 3 so we have a result there of roughly 0 .0 it's effectively 0 .05 basically if we wanted to round that to a reasonable 9 .5 number of decimal places.
04:06
And i'll double check the result there.
04:08
Oops, not p of 58.
04:09
We want p of 8.
04:10
So we can see we get the result, or the same result, rather...