suppose that you have 6 green cards and 5 yellow cards. the cards are well shuffled. you randomly draw two…

Explanation:

Step1: Calculate total number of cards

The total number of cards is $6 + 5=11$.

Step2: Calculate $P(G_1$ and $G_2)$

Since the draws are with replacement, $P(G_1)=\frac{6}{11}$ and $P(G_2)=\frac{6}{11}$. For independent - events, $P(G_1\cap G_2)=P(G_1)\times P(G_2)$. So $P(G_1\cap G_2)=\frac{6}{11}\times\frac{6}{11}=\frac{36}{121}\approx0.2975$.

Step3: Calculate $P(\text{At least 1 green})$

First, find $P(\text{no green})$. $P(\text{no green}) = P(\text{both yellow})$. Since $P(\text{yellow})=\frac{5}{11}$, $P(\text{both yellow})=\frac{5}{11}\times\frac{5}{11}=\frac{25}{121}$. Then $P(\text{at least 1 green}) = 1 - P(\text{no green})=1-\frac{25}{121}=\frac{96}{121}\approx0.7934$.

Step4: Calculate $P(G_2|G_1)$

Since the draws are with replacement, the probability that the second card is green given that the first card is green is the same as the probability of drawing a green card in a single draw. So $P(G_2|G_1)=\frac{6}{11}\approx0.5455$.

Step5: Determine independence

Two events $G_1$ and $G_2$ are independent if $P(G_2|G_1)=P(G_2)$. Here, $P(G_2|G_1)=\frac{6}{11}$ and $P(G_2)=\frac{6}{11}$, so they are independent events.

Answer:

a. $0.2975$ b. $0.7934$ c. $0.5455$ d. They are independent events