Question

find the odds of each of the following events.
(a) an event e with pr(e) = 4/9
(b) an event e with pr(e) = 0.25

(a) the odds are 4 to 5
(simplify your answer.)

(b) the odds are \\(\square\\) to \\(\square\\)
(simplify your answer.)

Explanation:

Step1: Recall the formula for odds

The formula to find the odds in favor of an event \( E \) is \( \text{Odds}(E)=\frac{\text{Pr}(E)}{1 - \text{Pr}(E)} \).

Step2: Convert the probability to a fraction

Given \( \text{Pr}(E) = 0.25=\frac{1}{4} \).

Step3: Calculate the probability of the complement

The probability of the complement of \( E \), \( \text{Pr}(E')=1-\text{Pr}(E)=1 - \frac{1}{4}=\frac{3}{4} \).

Step4: Find the odds

Using the formula for odds, \( \text{Odds}(E)=\frac{\text{Pr}(E)}{\text{Pr}(E')}=\frac{\frac{1}{4}}{\frac{3}{4}}=\frac{1}{3} \). So the odds are \( 1 \) to \( 3 \).

Answer:

The odds are \( 1 \) to \( 3 \)