91Ó°ÊÓ

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.

The tree represents the possibilities for the cookie selections.

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.

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

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

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

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

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

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

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

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