Question

if you roll one die, what is the probability of getting an even number or a multiple of 3?

Explanation:

Step1: Determine the sample space

When rolling a die, the sample space \(S = \{1, 2, 3, 4, 5, 6\}\), so \(n(S)=6\).

Step2: Define events

Let \(A\) be the event of getting an even number (\(A=\{2,4,6\}\), \(n(A) = 3\)) and \(B\) be the event of getting a multiple of \(3\) (\(B = \{3,6\}\), \(n(B)=2\)). The intersection \(A\cap B=\{6\}\), \(n(A\cap B) = 1\).

Step3: Use the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)

By the formula \(P(X)=\frac{n(X)}{n(S)}\), we have \(P(A)=\frac{3}{6}\), \(P(B)=\frac{2}{6}\), \(P(A\cap B)=\frac{1}{6}\). Then \(P(A\cup B)=\frac{3 + 2-1}{6}=\frac{4}{6}=\frac{2}{3}\).

Answer:

\(\frac{2}{3}\)