00:01
All right, we're told there are eight fruit trees, and we're told that there are 10 pine trees, and then we're told that there are 14 maple trees.
00:18
And we have people coming to select these, and we're going to find, for part one, for a, we're going to find the probability that out of four people, all four people pick the same.
00:31
So all four people, all four pick the same tree.
00:40
And we're going to assume it's independent.
00:43
The choices are independent of each other.
00:45
So before we go to begin, we're going to be using the n -chus -x -factorial, the combinations formula.
00:54
So that's given as n -factorial over x -factorial times n -minus x quantity factorial.
01:01
We'll be using this in some of these later exercises.
01:05
So we'll just go ahead and put that there.
01:07
But for the first part, we don't need to do that because, well, what's the probability that all four people will pick the fruit tree? well, there are eight, and then there are, there's eight out of a total of 32.
01:23
There's 32 total, right? 10 and 14 make 24, 24 and 8 is 32.
01:29
And then we have, after that, there are seven fruit trees left, but there are 31 to choose from total.
01:38
And then there's six out of 30, and then at that point it was five out of 29.
01:43
So this is the probability that all four pick the fruit trees.
01:47
All fruit trees.
01:53
Let's say four fruit trees.
01:54
That you multiply all that together.
01:56
And then the probability of four pine trees, that's going to be equal to, let's see, 10 out of 32 times 9 out of 31 times 8 at a 30, times 7 out of 29.
02:14
And then the probability of four maple trees.
02:17
Out of four people is equal to 14 out of 32 times 13 out of 31 times 12 out of 30 times 11 out of 29.
02:31
So the probability that all four pick the same will be the sum of all these.
02:35
Because there's no overlap here, these are all mutually exclusive, so we can add them up.
02:39
That's the probability that all four people will pick the same.
02:41
And it could be this one, it could be this one or this one, but it's the, it's this, the union of all these.
02:46
So it's the probability of four fruit trees plus the probability of four pine trees, plus the probability of four maple trees.
02:58
And when we add all that together, we get 0 .0356.
03:05
For part b, this is where we're going to get into our combinations.
03:10
We want to know the probability of the four people.
03:14
Three pick the pine, and one person picks.
03:19
A maple tree.
03:21
So let's think about the three pine trees first.
03:23
So there are 10 pine trees, and we're going to choose three of them.
03:27
And then the total ways we can get three trees out of this would be 32, choose three.
03:35
But then we want the one maple tree.
03:37
And so because these are independent, we can multiply the probabilities together.
03:40
And there are 14 maple trees, but we want one of them.
03:46
And then of that, there are 29 left to choose one from, right? because if you have 32 and you choose three of them, you would get 29, right? 32 minus 3 .2 is 29...