91影视

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, with replacement.

Let G1 = first card is green

Let G2 = second card is green

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 $P\left(G_2\right)=\frac{5}{8}$

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

The tree diagram of the situation is well-explained below

$P(G_{1}AND{G}_2)=\frac{5}{8}脳\frac{5}{8}=\frac{25}{64}$

Probability of at least one green

$$\begin{gathered} P(atleastonegreen)=P\left(GG\right)+P\left(GY\right)+P\left(YG\right) \\ P(atleastonegreen)=\frac{5}{8}脳\frac{5}{8}+\frac{5}{8}脳\frac{3}{8}+\frac{3}{8}脳\frac{5}{8} \\ P(atleastonegreen)=\frac{25}{64}+\frac{15}{64}+\frac{15}{64} \\ P(atleastonegreen)=\frac{55}{64} \end{gathered}$$

$$\begin{gathered} P\left(\overline{)G_1}{G}_2\right)=\frac{P(G_{1}AND{G}_2)}{P\left(G_2\right)} \\ P\left(\overline{)G_1}{G}_2\right)=\frac{{\displaystyle \frac{25}{64}}}{{\displaystyle \frac{5}{8}}}=\frac{5}{8} \end{gathered}$$

Yes, they are independent because the first card is placed back in the bag before the second card is drawn; the composition of cards in the bag remains the same from draw one to draw two.