Question

let a and b be mutually exclusive events with $p(a) = \frac{1}{4}$ and $p(b) = \frac{1}{5}$. what is $p(a \text{ or } b)$? write your answer as a fraction or decimal. do not round.

Explanation:

Step1: Recall the formula for mutually exclusive events

For mutually exclusive events \( A \) and \( B \), the probability of \( A \) or \( B \) is given by \( P(A \cup B)=P(A) + P(B) \) because \( P(A \cap B) = 0 \) (mutually exclusive events cannot occur simultaneously).

Step2: Substitute the given probabilities

We know that \( P(A)=\frac{1}{4} \) and \( P(B)=\frac{1}{5} \). So we need to add these two fractions. First, find a common denominator. The least common denominator of 4 and 5 is 20.

Convert \( \frac{1}{4} \) to a fraction with denominator 20: \( \frac{1}{4}=\frac{1\times5}{4\times5}=\frac{5}{20} \)

Convert \( \frac{1}{5} \) to a fraction with denominator 20: \( \frac{1}{5}=\frac{1\times4}{5\times4}=\frac{4}{20} \)

Now add the two fractions: \( P(A \cup B)=\frac{5}{20}+\frac{4}{20}=\frac{5 + 4}{20}=\frac{9}{20} \) (or as a decimal, \( \frac{9}{20}=0.45 \))

Answer:

\( \frac{9}{20} \) (or \( 0.45 \))