Question

Suppose that you randomly draw two cards, one at a time, without replacement.

G1 = first card is green

G2 = second card is green

a. Draw a tree diagram of the situation.

b. Find P(G1 AND G2).

c. Find P(at least one green).

d. Find P(G2|G1).

e. Are G2 and G1 independent events? Explain why or why not.

Short answer

a. Tree diagram

b. $\frac{20}{56}$

c.$\frac{42}{56}$

d. $\frac{5}{8}$

e. No

Step-by-step solution

Given information

Suppose that you have eight cards.

Five are green and three are yellow.

The cards are well shuffled.

Suppose that you randomly draw two cards, one at a time, without replacement.

Let G1 = first card is green

Let G2 = second card is green

Explanation (part a)

The probability of first card is green is given by $P\left(G_1\right)=\frac{5}{8}$

The probability of second card is green is given by $P\left(G_2\right)=\frac{4}{7}$

The probability of first card is yellow is given by$P\left(Y_1\right)=\frac{3}{8}$

The probability of second card is yellow is given by $P\left(Y_2\right)=\frac{2}{7}$

The tree diagram of the situation is well-explained below

Explanation (part b)

$$\begin{gathered} P(G_{1}AND{G}_2)=P\left(G_1\right).P\left(G_2\right) \\ P(G_{1}AND{G}_2)=\frac{5}{8}\times \frac{4}{7} \\ P(G_{1}AND{G}_2)=\frac{20}{56} \end{gathered}$$

Explanation (part c)

Probability of at least one green

$$\begin{gathered} P(atleastonegreen)=P\left(GG\right)+P\left(GY\right)+P\left(YG\right) \\ P(atleastonegreen)=\frac{5}{8}\times \frac{4}{7}+\frac{5}{8}\times \frac{2}{7}+\frac{3}{8}\times \frac{4}{7} \\ P(atleastonegreen)=\frac{20}{56}+\frac{10}{56}+\frac{12}{56} \\ P(atleastonegreen)=\frac{42}{56} \end{gathered}$$

Explanation (part d)

$$\begin{gathered} P\left(\overline{)G_1}{G}_2\right)=\frac{P(G_{1}AND{G}_2)}{P\left(G_2\right)} \\ P\left(\overline{)G_1}{G}_2\right)=\frac{{\displaystyle \frac{20}{56}}}{{\displaystyle \frac{4}{7}}}=\frac{5}{8} \end{gathered}$$

Explanation (part e)

No, they are dependent because the first card is drawn in the bag before the second card is drawn; the composition of cards in the bag doesn't remain the same from draw one to draw two.