You roll a pair of fair six-sided dice. The sample space of 36 possible outcomes is shown below. Based on this information, answer the following questions.

What is $P(A)$, the probability that the first die is a 5 ? $\qquad$
What is $P(B)$, the probability that the second die is a 3 ? $\qquad$
What is $P(A$ and $B)$, the probability that the first die is a 5 and the second die is a 3 ? $\square$
What is $P(B \mid A)$, the conditional probability that the second die is a 3 given that the first die is a 5 ? $\square$
Is $P(B \mid A)=P(B)$ ? Are the events $A$ and $B$ independent?
Choose all answers that apply:
A Yes, $P(B \mid A)=P(B)$.
B No, $P(B \mid A)
eq P(B)$.
c. Yes, events $A$ and $B$ are independent events.

Do, events $A$ and $B$ are not independent events.

Question

You roll a pair of fair six-sided dice. The sample space of 36 possible outcomes is shown below. Based on this information, answer the following questions.

What is $P(A)$, the probability that the first die is a 5 ? $\qquad$
What is $P(B)$, the probability that the second die is a 3 ? $\qquad$
What is $P(A$ and $B)$, the probability that the first die is a 5 and the second die is a 3 ? $\square$
What is $P(B \mid A)$, the conditional probability that the second die is a 3 given that the first die is a 5 ? $\square$
Is $P(B \mid A)=P(B)$ ? Are the events $A$ and $B$ independent?
Choose all answers that apply:
A Yes, $P(B \mid A)=P(B)$.
B No, $P(B \mid A) 
eq P(B)$.
c. Yes, events $A$ and $B$ are independent events.

Do, events $A$ and $B$ are not independent events.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find $P(A)$, the probability that the first die is a 5, we note that there are 6 possible outcomes for the first die, and only one of these outcomes is a 5. Therefore, $P(A) = \frac{1}{6}$. ‖ ‖ step 2 ⋮ To find $P(B)$, the probability that the second die is a 3, we note that there are 6 possible outcomes for the second die, and only one of these outcomes is a 3. Therefore, $P(B) = \frac{1}{6}$. ‖ ‖ step 3 ⋮ To find $P(A ext{ and } B)$, the probability that the first die is a 5 and the second die is a 3, we note that there is only one outcome where the first die is a 5 and the second die is a 3 out of the 36 possible outcomes. Therefore, $P(A ext{ and } B) = \frac{1}{36}$. ‖ ‖ step 4 ⋮ To find $P(B \mid A)$, the conditional probability that the second die is a 3 given that the first die is a 5, we use the formula for conditional probability: $P(B \mid A) = \frac{P(A ext{ and } B)}{P(A)} = \frac{\frac{1}{36}}{\frac{1}{6}} = \frac{1}{6}$. ‖ ‖ step 5 ⋮ To determine if $P(B \mid A) = P(B)$, we compare the two probabilities. We have $P(B \mid A) = \frac{1}{6}$ and $P(B) = \frac{1}{6}$. Since $P(B \mid A) = P(B)$, the events $A$ and $B$ are independent. ‖ ∻Answer∻ ***A*** ∻Key Concept∻ ⚹Independence of Events⚹ ∻Explanation∻ ⚹Two events $A$ and $B$ are independent if and only if $P(B \mid A) = P(B)$.⚹◊What is the formula for conditional probability $P(B \mid A)$ in this scenario?⍭ Generate me a similar question◊

Sia