00:01
We're looking at a town council.
00:03
It has five democrats and six republicans.
00:08
On a, we are picking two people at random.
00:11
One will be president, one will be vice president.
00:14
What is the probability they are both democrats? so we are picking two people without replacement and order matters.
00:24
So this is going to be permutations.
00:28
So in permutations, the order in which you select them matters.
00:33
In combinations, it doesn't matter.
00:35
We'll come to that in part b.
00:37
And the formula for permutations, n -pr, is in factorial over n minus r, factorial.
00:47
So this tells you the number of ways you can pick that many people from n members.
00:53
So what i'm going to do is make this big fraction.
00:56
I'm going to have all the ways of picking them on the denominator, and on the numerator i'm going to have the ways that meet my crameter.
01:06
Criteria.
01:14
So each way of choosing them, because it's completely at random, they're all equally likely to have.
01:19
So each way of choosing them is one over the number of ways of picking them.
01:23
And if i just have the number of ways that have both democrats, i'll get that.
01:29
So how do i get all of the ways? well, i've got 11 people and i just need to pick two of them, all the matters, so limitations.
01:38
How many ways can i pick both democrats? well, there's only five democrats, and and i'm picking two of them.
01:45
So it's 5p2 over 11p2.
01:49
So 5p2 is 5 factorial divided by 3 factorial, it's 20.
01:58
So there are 20 ways to pick two democrats for these two positions.
02:07
And then 11 factorial, divided by 9 factorial, is 110.
02:15
There are 110 ways to pick these two people just from the entire group.
02:21
Which gives us the answer 0 .1818 to four decimal places.
02:29
So that's part a.
02:31
Part b is going to be pretty similar, but we're looking at combinations now.
02:36
Because now the order doesn't matter, we're just picking a committee...