Question

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

(a) the odds are 5 to 2
(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 \( \text{Odds}(E)=\frac{\text{Pr}(E)}{1 - \text{Pr}(E)} \), and the odds are expressed as \( \text{Pr}(E):(1 - \text{Pr}(E)) \).

Step2: Solve part (a)

Given \( \text{Pr}(E)=\frac{5}{7} \), first find \( 1-\text{Pr}(E) \).
\( 1-\text{Pr}(E)=1 - \frac{5}{7}=\frac{2}{7} \)
Then the odds are \( \frac{\text{Pr}(E)}{1 - \text{Pr}(E)}=\frac{\frac{5}{7}}{\frac{2}{7}}=\frac{5}{2} \), so the odds are \( 5 \) to \( 2 \).

Step3: Solve part (b)

Given \( \text{Pr}(E) = 0.6=\frac{3}{5} \), find \( 1-\text{Pr}(E) \).
\( 1-\text{Pr}(E)=1 - 0.6 = 0.4=\frac{2}{5} \)
Then the odds are \( \frac{\text{Pr}(E)}{1 - \text{Pr}(E)}=\frac{\frac{3}{5}}{\frac{2}{5}}=\frac{3}{2} \), so the odds are \( 3 \) to \( 2 \).

Answer:

(a) The odds are \( 5 \) to \( 2 \)
(b) The odds are \( 3 \) to \( 2 \)