Question

a single, six - sided die is rolled. find the probability of rolling an odd number or a number less than 4.
the probability is \square. (type an integer or a fraction. simplify your answer.)

Explanation:

Step1: Determine sample space

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

Step2: Define events

Let \(A\) be the event of rolling an odd number. Then \(A=\{1,3,5\}\), and \(n(A)=3\). Let \(B\) be the event of rolling a number less than \(4\). Then \(B = \{1,2,3\}\), and \(n(B)=3\). The intersection \(A\cap B=\{1,3\}\), and \(n(A\cap B) = 2\).

Step3: Use the addition rule

The addition rule for 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 \(P(A\cap B)=\frac{n(A\cap B)}{n(S)}\), we have \(P(A\cup B)=\frac{n(A)+n(B)-n(A\cap B)}{n(S)}\). Substitute \(n(A) = 3\), \(n(B)=3\), \(n(A\cap B)=2\) and \(n(S)=6\) into the formula: \(P(A\cup B)=\frac{3 + 3-2}{6}=\frac{4}{6}=\frac{2}{3}\).

Answer:

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