Questions: An event X will occur with probability 0.4. The conditional probability that event Y will occur given that X occurs is 0.8. What is the probability that both X and Y will occur? A 0.40 B 0.12 C 0.32 D 0.50 E 0.20

An event X will occur with probability 0.4. The conditional probability that event Y will occur given that X occurs is 0.8.

What is the probability that both X and Y will occur?

A 0.40

B 0.12

C 0.32

D 0.50

E 0.20

Transcript text: An event $X$ will occur with probability 0.4 . The conditional probability that event $Y$ will occur given that $X$ occurs is 0.8 . What is the probability that both $X$ and $Y$ will occur? A 0.40 B $\quad 0.12$ C $\quad 0.32$ D 0.50 E $\quad 0.20$

Solution

Solution Steps

Step 1: Identify the given probabilities

We are given:

$ P(X) = 0.4 $ (the probability that event $ X $ occurs).
$ P(Y|X) = 0.8 $ (the conditional probability that event $ Y $ occurs given that $ X $ occurs).

Step 2: Use the definition of conditional probability

The conditional probability $ P(Y|X) $ is defined as: \[ P(Y|X) = \frac{P(X \cap Y)}{P(X)} \] where $ P(X \cap Y) $ is the probability that both $ X $ and $ Y $ occur.

Step 3: Solve for $ P(X \cap Y) $

Rearrange the formula to solve for $ P(X \cap Y) $: \[ P(X \cap Y) = P(Y|X) \cdot P(X) \] Substitute the given values: \[ P(X \cap Y) = 0.8 \cdot 0.4 = 0.32 \]

Final Answer

$\boxed{0.32}$

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