Question

a single, six - sided die is rolled. find the probability of rolling an even number or a number less than 3. the probability is (type an integer or a fraction. simplify your answer.)

Explanation:

Step1: Determine the sample space

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

Step2: Define the events

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

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 \(P(A)=\frac{n(A)}{n(S)}\), \(P(B)=\frac{n(B)}{n(S)}\) and \(P(A\cap B)=\frac{n(A\cap B)}{n(S)}\), we have:

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Answer:

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