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 \\(\square\\) to \\(\square\\).
(simplify your answer.)

Explanation:

Step1: Recall the formula for odds

The formula for the odds in favor of an event \( E \) is \( \text{Odds}(E)=\frac{\text{Pr}(E)}{1 - \text{Pr}(E)} \). First, we need to find \( 1-\text{Pr}(E) \) when \( \text{Pr}(E)=\frac{4}{9} \).

Step2: Calculate \( 1 - \text{Pr}(E) \)

\( 1-\text{Pr}(E)=1-\frac{4}{9}=\frac{9 - 4}{9}=\frac{5}{9} \).

Step3: Calculate the odds

Now, the odds in favor of \( E \) is \( \frac{\text{Pr}(E)}{1 - \text{Pr}(E)}=\frac{\frac{4}{9}}{\frac{5}{9}} \). When dividing fractions, we multiply by the reciprocal, so \( \frac{4}{9}\times\frac{9}{5}=\frac{4}{5} \). This means the odds are \( 4 \) to \( 5 \).

Answer:

\( 4 \) to \( 5 \)