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 and b)
p(a and b) = \boxed{}
(simplify your answer.)

Explanation:

Step1: Define Events A and B

Event A: Selected number is even. So \( A = \{2, 4, 6, 8, 10, 12, 14\} \), \( n(A)=7 \).
Event B: Selected number is a multiple of 4. So \( B = \{4, 8, 12\} \), \( n(B)=3 \).

Step2: Find \( A \cap B \)

\( A \cap B \) is numbers that are even and multiples of 4. So \( A \cap B = \{4, 8, 12\} \), \( n(A \cap B)=3 \).

Step3: Calculate \( P(A \text{ and } B) \)

Sample space \( S \) has \( n(S)=15 \) (since \( S = \{1,2,\dots,15\} \)).
Probability formula: \( P(A \text{ and } B) = \frac{n(A \cap B)}{n(S)} \).
Substitute values: \( \frac{3}{15} = \frac{1}{5} \).

Answer:

\(\frac{1}{5}\)