Question

stacy is choosing an entree and a dessert from a restaurant menu. let a be the event that she orders a sandwich for her entree. let b be the event that she orders something chocolate for dessert. stacys choice of dessert is independent of her choice of entree. choose two probabilities that must be equal. a. the probability stacy orders a chocolate dessert b. the probability stacy orders a sandwich or a chocolate dessert c. the probability stacy orders a sandwich and a chocolate dessert d. the probability stacy orders a chocolate dessert if she has already ordered a sandwich e. the probability stacy orders a sandwich if she has already ordered a chocolate dessert

Explanation:

Step1: Recall the definition of independent events

If two events $A$ and $B$ are independent, then $P(B|A)=P(B)$ and $P(A|B) = P(A)$. Here, event $A$ is ordering a sandwich for entree and event $B$ is ordering something chocolate for dessert, and they are independent.

Step2: Analyze each option

• Option A: $P(B)$ is the probability Stacy orders a chocolate dessert.
• Option B: $P(A\cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B)-P(A)P(B)$ (since $A$ and $B$ are independent).
• Option C: $P(A\cap B)=P(A)P(B)$ (by the formula for independent - events).
• Option D: $P(B|A)=\frac{P(A\cap B)}{P(A)}$. Since $P(A\cap B)=P(A)P(B)$ for independent events, $P(B|A) = P(B)$.
• Option E: $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Since $P(A\cap B)=P(A)P(B)$ for independent events, $P(A|B)=P(A)$.

Since $P(B|A) = P(B)$ for independent events $A$ and $B$, the probability Stacy orders a chocolate dessert (Option A) is equal to the probability Stacy orders a chocolate dessert if she has already ordered a sandwich (Option D).

Answer:

A. the probability Stacy orders a chocolate dessert
D. the probability Stacy orders a chocolate dessert if she has already ordered a sandwich