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: Identify sample space and events

A six - sided die has sample space \(S=\{1,2,3,4,5,6\}\), so \(n(S) = 6\). Let \(A\) be the event of rolling an odd number, so \(A=\{1,3,5\}\), \(n(A)=3\). Let \(B\) be the event of rolling a number less than 4, so \(B = \{1,2,3\}\), \(n(B)=3\).

Step2: Find \(A\cap B\)

The intersection \(A\cap B\) is the set of numbers that are both odd and less than 4. So \(A\cap B=\{1,3\}\), \(n(A\cap B) = 2\).

Step3: Use the formula for \(P(A\cup B)\)

The formula for the probability of the union of two events is \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Since \(P(A)=\frac{n(A)}{n(S)}=\frac{3}{6}\), \(P(B)=\frac{n(B)}{n(S)}=\frac{3}{6}\), and \(P(A\cap B)=\frac{n(A\cap B)}{n(S)}=\frac{2}{6}\).

Substitute the values:
\(P(A\cup B)=\frac{3}{6}+\frac{3}{6}-\frac{2}{6}=\frac{3 + 3-2}{6}=\frac{4}{6}=\frac{2}{3}\)

Answer:

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