Question

for the experiment of rolling a single fair die, find the probability of being even or prime. (hint: recall that 1 is neither prime nor composite.) the probability is (simplify your answer.)

Explanation:

Step1: Define sample space and events

The sample space when rolling a die is $S = \{1, 2, 3, 4, 5, 6\}$, $n(S)=6$. Let $A$ be the event of getting an even - number, $A=\{2,4,6\}$, $n(A) = 3$. Let $B$ be the event of getting a prime number, $B=\{2,3,5\}$, $n(B)=3$. And $A\cap B=\{2\}$, $n(A\cap B) = 1$.

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 $P(A\cap B)=\frac{n(A\cap B)}{n(S)}$, we have $P(A\cup B)=\frac{n(A)}{n(S)}+\frac{n(B)}{n(S)}-\frac{n(A\cap B)}{n(S)}$.

Step3: Substitute the values

Substitute $n(A) = 3$, $n(B)=3$, $n(A\cap B)=1$ and $n(S)=6$ into the formula: $P(A\cup B)=\frac{3}{6}+\frac{3}{6}-\frac{1}{6}=\frac{3 + 3-1}{6}=\frac{5}{6}$.

Answer:

$\frac{5}{6}$