Question

One of the two events must occur. The chance of one is \( \frac{2}{3} \) of the other, then odds in favor of the other are

• A. \( 3:2 \)
• B. \( 1:3 \)
• C. \( 3:1 \)
• D. \( 2:3 \)

Hint:

Odds are often written as "successes:failures," where failures are the complement of the event happening.

Answer 1

The Correct Option is A

Let the probability of one event occurring be \( P(A) \) and the probability of the other event occurring be \( P(B) \).

Given that the chance of one event is \( \frac{2}{3} \) of the other, we can express this as:

\[ P(A) = \frac{2}{3} P(B) \]

Since one of the events must occur, the sum of their probabilities is 1:

\[ P(A) + P(B) = 1 \]

Substitute \( P(A) = \frac{2}{3} P(B) \) into the equation:

\[ \frac{2}{3} P(B) + P(B) = 1 \]

Factor out \( P(B) \):

\[ P(B) \left( \frac{2}{3} + 1 \right) = 1 \]

Now simplify the expression:

\[ P(B) \left( \frac{5}{3} \right) = 1 \]

Solve for \( P(B) \):

\[ P(B) = \frac{3}{5} \]

Now substitute \( P(B) = \frac{3}{5} \) into \( P(A) = \frac{2}{3} P(B) \):

\[ P(A) = \frac{2}{3} \times \frac{3}{5} = \frac{2}{5} \]

The odds in favor of the other event occurring is the ratio of \( P(B) \) to \( P(A) \):

\[ \text{Odds in favor of the other event} = \frac{P(B)}{P(A)} = \frac{\frac{3}{5}}{\frac{2}{5}} = \frac{3}{2} \]

The odds in favor of the other event are 3 : 2, which corresponds to option (A).