Question

The odds in favor of winning a game can be found by computing the ratio of the probability of winning to the probability of not winning. If the probability that Pat will win a game is \(\frac{4}{9}\), what are the odds that Pat will win the game?

• A. 4 to 5
• B. 4 to 9
• C. 5 to 4
• D. 5 to 9
• E. 9 to 5

Hint:

Don't confuse probability with odds. If the probability of winning is \(\frac{a}{b}\), it means 'a' successes in 'b' total trials. The odds are a ratio of successes to failures. In this case, 4 successes and \(9-4=5\) failures, so the odds are 4 to 5.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The question asks for the "odds in favor" of an event, and provides the definition. We need to use the given probability of winning to find the probability of not winning, and then form the specified ratio.
Step 2: Key Formula or Approach:
Odds in favor = \(\frac{P(\text{winning})}{P(\text{not winning})}\)
Also, the probability of an event not happening is 1 minus the probability of it happening:
\(P(\text{not winning}) = 1 - P(\text{winning})\)
Step 3: Detailed Explanation:
1. Identify the probability of winning.
We are given \(P(\text{winning}) = \frac{4}{9}\).
2. Calculate the probability of not winning.
\[ P(\text{not winning}) = 1 - \frac{4}{9} = \frac{9}{9} - \frac{4}{9} = \frac{5}{9} \]
3. Compute the ratio for the odds in favor.
\[ \text{Odds in favor} = \frac{P(\text{winning})}{P(\text{not winning})} = \frac{4/9}{5/9} \]
To divide by a fraction, we multiply by its reciprocal:
\[ \text{Odds in favor} = \frac{4}{9} \times \frac{9}{5} = \frac{4}{5} \]
This ratio is expressed as "4 to 5".
Step 4: Final Answer:
The odds that Pat will win the game are 4 to 5.