Question

A box of cookies contains three chocolate and seven butter cookies. Miguel randomly selects a cookie and eats it. Then he randomly selects another cookie and eats it. (How many cookies did he take?)

a. Draw the tree that represents the possibilities for the cookie selections. Write the probabilities along each branch of the tree.

b. Are the probabilities for the flavor of the SECOND cookie that Miguel selects independent of his first selection? Explain.

c. For each complete path through the tree, write the event it represents and find the probabilities.

d. Let S be the event that both cookies selected were the same flavor. Find P(S).

e. Let T be the event that the cookies selected were different flavors. Find P(T) by two different methods: by using the complement rule and by using the branches of the tree. Your answers should be the same with both methods.

f. Let U be the event that the second cookie selected is a butter cookie. Find P(U).

Short answer

a.

b. no solution, because the SECOND cookie probability selection will be different depending on which cookie is selected first.

c. $P(BANDB)=\frac{42}{90};P(BANDC)=\frac{21}{90};P(CANDC)=\frac{6}{90}$

d. $P\left(S\right)=\frac{21}{81}$

e. $P\left(T\right)=\frac{60}{81}$

d.$P\left(U\right)=\frac{42}{90}$

Step-by-step solution

Given information

A box of cookies contains three chocolate and seven butter cookies.

Miguel randomly selects a cookie and eats it. Then he randomly selects another cookie and eats it.

Part (a) Step 1: Explanation 

The tree represents the possibilities for the cookie selections.

Part (b) Step 1: Explanation 

the probabilities for the flavor of the SECOND cookie that Miguel selects independent of his first selection give no solution. because the SECOND cookie probability selection will be different depending on which cookie is selected first.

Part (c) Step 1: Explanation 

For each complete path through the tree, write the event it represents and find the probabilities:

$$\begin{gathered} P(BANDB)=\frac{7}{10}\times \frac{6}{9}=\frac{42}{90} \\ P(BANDC)=\frac{7}{10}\times \frac{3}{9}=\frac{21}{90} \\ P(CANDB)=\frac{3}{10}\times \frac{7}{9}=\frac{21}{90} \\ P(CANDC)=\frac{3}{10}\times \frac{2}{9}=\frac{6}{90} \end{gathered}$$

Part (d) Step 1: Explanation 

Let S be the event that both cookies selected were the same flavor. The probability of the event is given by

$P\left(S\right)=P(CANDB)=P\left(C\right).P\left(B\right)=\frac{3}{9}\times \frac{7}{9}=\frac{21}{81}$

Part (e) Step 1: Explanation 

Let T be the event that the cookies selected were different flavors. The probability of the event is given by the tree diagram.

$$\begin{gathered} P\left(T\right)=P(CORB)=P\left(C\right)+P\left(B\right)-P\left(S\right)=\frac{3}{10}+\frac{7}{10}-\frac{21}{81} \\ P\left(T\right)=\frac{10}{10}-\frac{21}{81}=\frac{810-210}{810}=\frac{600}{810}=\frac{60}{81} \end{gathered}$$

The probability of the event is given by complement method as

$P\left(T\right)=P\left(S^{\text{'}}\right)=1-P\left(S\right)=1-\frac{21}{81}=\frac{81-21}{81}=\frac{60}{81}$

Part (f) Step 1: Explanation 

Let U be the event that the second cookie selected is a butter cookie. The probability of the event is given by

$P\left(U\right)=P(BANDB)=\frac{42}{90}$