Question

the table gives a set of outcomes and their probabilities. let a be the event \the outcome is prime\. let b be the event \the outcome is a divisor of 4\. find ( p(a|b) ).

outcome	probability
2	0.1
3	0.2
4	0.4

Explanation:

Step1: Identify events A and B

Event B: outcomes are divisors of 4. Divisors of 4 are 1, 2, 4. So \( B = \{1, 2, 4\} \).
Event A: outcomes are prime. Primes in the outcomes are 2, 3. So \( A \cap B = \{2\} \) (since 2 is in both A and B).

Step2: Calculate \( P(B) \) and \( P(A \cap B) \)

\( P(B) = P(1) + P(2) + P(4) = 0.3 + 0.1 + 0.4 = 0.8 \).
\( P(A \cap B) = P(2) = 0.1 \).

Step3: Apply conditional probability formula

The formula for conditional probability is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \).
Substitute the values: \( P(A|B) = \frac{0.1}{0.8} = \frac{1}{8} = 0.125 \).

Answer:

\( 0.125 \)