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(c|a) p(c|a) = (simplify your answer. type an integer or a fraction.)

Explanation:

Step1: Find elements in event A

The sample - space \(S=\{1,2,\cdots,15\}\). The set of even numbers in \(S\) is \(A = \{2,4,6,8,10,12,14\}\), so \(n(A)=7\).

Step2: Find elements in \(A\cap C\)

The set of prime numbers in \(S\) is \(C=\{2,3,5,7,11,13\}\). The intersection of \(A\) and \(C\) is \(A\cap C = \{2\}\), so \(n(A\cap C)=1\).

Step3: Use the formula for conditional probability

The formula for conditional probability is \(P(C|A)=\frac{P(A\cap C)}{P(A)}\). Since \(P(A\cap C)=\frac{n(A\cap C)}{n(S)}\) and \(P(A)=\frac{n(A)}{n(S)}\), then \(P(C|A)=\frac{n(A\cap C)}{n(A)}\).
Substituting \(n(A\cap C) = 1\) and \(n(A)=7\) into the formula, we get \(P(C|A)=\frac{1}{7}\).

Answer:

\(\frac{1}{7}\)