The odds of an event are defined as the ratio of the probability that the event occurs to the probability that it does not occur. If $P(E)$ is the probability of the event occurring, then the odds of the event $E$ are given by the formula:

$\text{Odds of } E = \frac{P(E)}{1 - P(E)}$

Let's calculate the odds for both events:

(a) Event $E$ with $P(E) = \frac{2}{9}$

$\text{Odds of } E = \frac{\frac{2}{9}}{1 - \frac{2}{9}} = \frac{\frac{2}{9}}{\frac{7}{9}} = \frac{2}{7}$ So, the odds of event $E$ are $2:7$.

(b) Event $E$ with $P(E) = 0.40$

$\text{Odds of } E = \frac{0.40}{1 - 0.40} = \frac{0.40}{0.60} = \frac{2}{3}$ So, the odds of event $E$ are $2:3$.

Summary:

• (a) The odds of event $E$ with $P(E) = \frac{2}{9}$ are $2:7$.
• (b) The odds of event $E$ with $P(E) = 0.40$ are $2:3$.

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would you calculate the odds if the probability of an event is $\frac{1}{3}$?
2. What is the relationship between odds and probability?
3. How would the odds change if $P(E) = 0.75$?
4. How do the odds differ when $P(E) = 0.5$?
5. Can odds be greater than 1? What does it mean when they are?

Tip: Odds greater than 1 indicate that the event is more likely to occur than not.