Question

the table gives a set of outcomes and their probabilities. let a be the event \the outcome is divisible by 3\. let b be the event \the outcome is prime\. find p(a|b).

outcome	probability
2	0.2
3	0.3
4	0.2

Explanation:

Step1: Recall the formula for conditional probability

The formula for conditional probability is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \).

Step2: Identify the prime numbers (event B)

Prime numbers in the outcomes (1, 2, 3, 4) are 2 and 3. So the outcomes in event B are 2 and 3.

Step3: Calculate \( P(B) \)

\( P(B) \) is the sum of probabilities of outcomes in B. So \( P(B)=P(2)+P(3)=0.2 + 0.3 = 0.5 \).

Step4: Identify the outcomes in \( A \cap B \)

Event A is outcomes divisible by 3, which is 3. Event B is prime numbers (2, 3). So \( A \cap B \) is the outcome 3 (since 3 is divisible by 3 and prime).

Step5: Calculate \( P(A \cap B) \)

\( P(A \cap B)=P(3)=0.3 \).

Step6: Calculate \( P(A|B) \)

Using the formula \( P(A|B)=\frac{P(A \cap B)}{P(B)}=\frac{0.3}{0.5} = 0.6 \).

Answer:

\( 0.6 \)