00:01
A box of cookies contains three chocolate and seven butter cookies.
00:05
Miguel randomly selects a cookie and eats it.
00:08
Then he randomly selects another cookie and eats it.
00:11
So he basically is taking two different cookies, and we want to draw a tree that represents the possibilities for the cookie selections, and we're going to write the probabilities along the branch of each tree.
00:25
So for a, we're going to draw a tree diagram.
00:29
So there's two possibilities.
00:31
He could either choose a chocolate cookie or he could choose a butter cookie.
00:38
We know that there are seven butter cookies and three chocolate so we have a total of ten cookies and we're going to write that as chocolate is three over ten and butter will be seven over ten.
00:56
Then he reaches in and picks out a second cookie.
00:58
Well on the second cookie you've got to remember that there's only nine cookies left in the box.
01:03
So if he chooses a chocolate or if you choose as a butter there's going to be nine cookies remaining well if he chooses a chocolate after already choosing a chocolate there's only two chocolates remaining out of nine cookies but if he chooses a butter there are still seven butter cookies out of the nine on the bottom of the table we have the same situation after selecting a butter cookie he selects one more he could select chocolate or butter there's still nine cookies available to him and and because the first selection was of butter in this case, there's still three chocolate remaining, and there's only six butter out of the nine cookies.
01:49
So part a or tree diagram is complete, in which he selects one cookie, then goes back and selects another cookie.
01:56
In question b, we're asked, are there probabilities for the flavor of the second cookie that miguel selects independent of his first selection? well, in this case, because you take a cookie in the first selection and then a second cookie and the second selection, the probability of selecting either a chocolate or a butter is going to be dependent on what you did in the first cookie selection.
02:24
So they're not dependent.
02:27
So in words, it would look like this.
02:29
No, because the second cookie selection's probability will be different depending on which cookie was selected first.
02:35
You can see this from the tree diagram.
02:37
If we select a chocolate cookie first, then the probability of a second chocolate cookie being two nights.
02:45
But if we have to select it a butter cookie first, the probability of a chocolate would have been three nights.
02:48
You can see that the second row of branches are different on each one, and that was all dependent on the first cookie selection.
02:58
For part c, for each complete path through the tree, write the event that it represents and find the probabilities.
03:09
So if you look back at a, if we make paths through the tree, we could go a chocolate cookie as our first selection and then we could go to a chocolate cookie as our second selection.
03:25
Or we could branch that off and have a butter cookie as our second selection.
03:30
So there's two possibilities there.
03:33
On the bottom side of the tree we could take a butter cookie first, then our second cookie could be chocolate or we could stick to the butter cookie and have a second butter cookie.
03:45
So there's four different possibilities here.
03:48
So when we write these out, we have the probability of butter and butter, butter and chocolate, chocolate and butter, and then we have chocolate and chocolate.
04:17
Now in order to figure out each of these probabilities, if we go down the branches of the tree diagram and multiply each fraction as we come to them, so, for example, for the butter and the butter, we start by going down the tree.
04:31
The first butter cookie has a 7 and 10 chance of being selected.
04:36
We're going to multiply that by the second branch of a butter as well, which has a 6 out of 9 chance of being selected.
04:45
And we get 42 .90ths, which reduces to approximately 0 .47.
04:57
And we can do this for each of the remaining situations.
05:00
So the probability of a butter and then a chocolate will end up giving you so seven tenths then if your second cookie is a chocolate you have a three out of nine chance which gives you 21 over 90 and that is approximately point two three for the chocolate then the butter it would be three tenths then you have a seven to nine chance for a butter again.
05:41
Gives you the exact same answer from above.
05:46
21 over 90, which is approximately 0 .23...