Question

for the experiment of rolling a single fair die, find the probability of being odd or divisible by 3. the probability is (simplify your answer.)

Explanation:

Step1: Determine sample space and events

Sample space \(S=\{1,2,3,4,5,6\}\), \(n(S) = 6\).
Let \(A\) be the event of getting an odd number: \(A=\{1,3,5\}\), \(n(A)=3\).
Let \(B\) be the event of getting a number divisible by \(3\): \(B = \{3,6\}\), \(n(B)=2\).
\(A\cap B=\{3\}\), \(n(A\cap B) = 1\).

Step2: Use the addition - rule of probability

The addition rule of probability is \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\).
Since \(P(A)=\frac{n(A)}{n(S)}\), \(P(B)=\frac{n(B)}{n(S)}\) and \(P(A\cap B)=\frac{n(A\cap B)}{n(S)}\).
Substitute the values: \(P(A)=\frac{3}{6}\), \(P(B)=\frac{2}{6}\), \(P(A\cap B)=\frac{1}{6}\).
\(P(A\cup B)=\frac{3 + 2-1}{6}\).

Answer:

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