Question

from the sample space s = {1,2,3,4,...,15}, a single number is to be selected at random. given the following events, find the indicated probability.
a: the selected number is even.
b: the selected number is a multiple of 4.
c: the selected number is a prime number.
p(a|c)
p(a|c) = \square
(simplify your answer.)

Explanation:

Step1: Define Events

Event \( A \): Selected number is even.
Event \( C \): Selected number is prime.
Sample space \( S = \{1,2,3,4,\dots,15\} \), so \( n(S) = 15 \).

Step2: Find \( n(C) \) (Prime numbers in \( S \))

Prime numbers in \( S \): \( 2,3,5,7,11,13 \). Thus, \( n(C) = 6 \).

Step3: Find \( n(A \cap C) \) (Even and prime numbers in \( S \))

Only \( 2 \) is even and prime. So \( n(A \cap C) = 1 \).

Step4: Apply Conditional Probability Formula

Conditional probability: \( P(A|C) = \frac{P(A \cap C)}{P(C)} = \frac{n(A \cap C)}{n(C)} \) (since \( P(C) = \frac{n(C)}{n(S)} \) and \( P(A \cap C) = \frac{n(A \cap C)}{n(S)} \), the \( n(S) \) cancels).

Substitute \( n(A \cap C) = 1 \) and \( n(C) = 6 \):
\( P(A|C) = \frac{1}{6} \).

Answer:

\(\frac{1}{6}\)