Question

if you roll one die, what is the probability of getting an odd number or a 4? are the events inclusive or mutually exclusive?

Explanation:

Step1: Determine sample space

A die has 6 faces: \(S = \{1,2,3,4,5,6\}\), \(n(S)=6\).

Step2: Define events

Let \(A\) be getting an odd number: \(A=\{1,3,5\}\), \(n(A) = 3\). Let \(B\) be getting a 4: \(B = \{4\}\), \(n(B)=1\).

Step3: Check mutual exclusivity

\(A\cap B=\varnothing\) (no common elements). So events are mutually - exclusive.

Step4: Calculate probability

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

Answer:

The probability is \(\frac{2}{3}\) and the events are mutually exclusive.