Question

what is the probability of either event occurring when you flip one coin and roll one die? event a: flipping a heads event b: rolling a 2 express your answer as a simplified fraction. ____

Explanation:

Step1: Calculate \(P(A)\), \(P(B)\) and \(P(A\cap B)\)

The probability of flipping a head \(P(A)=\frac{1}{2}\) (since there are 2 possible outcomes for a coin - head or tail).
The probability of rolling a 2 on a die \(P(B)=\frac{1}{6}\) (since there are 6 possible outcomes for a die - 1, 2, 3, 4, 5, 6).
Since flipping a coin and rolling a die are independent events, \(P(A\cap B)=P(A)\times P(B)=\frac{1}{2}\times\frac{1}{6}=\frac{1}{12}\).

Step2: Use the formula for the probability of the union of two events

The formula for \(P(A\cup B)\) is \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\).
Substitute the values: \(P(A\cup B)=\frac{1}{2}+\frac{1}{6}-\frac{1}{12}\).
First, find a common denominator, which is 12.
\(\frac{1}{2}=\frac{6}{12}\), \(\frac{1}{6}=\frac{2}{12}\).
Then \(P(A\cup B)=\frac{6 + 2-1}{12}=\frac{7}{12}\).

Answer:

\(\frac{7}{12}\)