Question

find the odds of each of the following events
(a) an event e with pr(e) = 5/7
(b) an event e with pr(e) = 0.6

Explanation:

Step1: Recall the formula for odds

The odds in favor of an event \( E \) is given by the ratio of the probability of \( E \) occurring to the probability of \( E \) not occurring. If \( \text{Pr}(E)=p \), then the probability of \( E \) not occurring, \( \text{Pr}(E') = 1 - p \). The odds in favor of \( E \) is \( \frac{p}{1 - p} \).

Step2: Solve part (a)

Given \( \text{Pr}(E)=\frac{6}{7} \), then \( \text{Pr}(E')=1-\frac{6}{7}=\frac{1}{7} \).
The odds in favor of \( E \) is \( \frac{\frac{6}{7}}{\frac{1}{7}}=\frac{6}{7}\times\frac{7}{1} = 6:1 \).

Step3: Solve part (b)

Given \( \text{Pr}(E) = 0.6=\frac{3}{5} \), then \( \text{Pr}(E')=1 - 0.6 = 0.4=\frac{2}{5} \).
The odds in favor of \( E \) is \( \frac{\frac{3}{5}}{\frac{2}{5}}=\frac{3}{5}\times\frac{5}{2}=\frac{3}{2}=3:2 \).

Answer:

(a) The odds of event \( E \) is \( 6:1 \)
(b) The odds of event \( E \) is \( 3:2 \)