There are two Porsches in a road race in Italy, and a

reporter feels that the odds against their winning are 3

to 1 and 5 to 3. To be consistent (see Exercise 82), what

odds should the reporter assign to the event that either

car will win?

To solve this problem, we need to find the combined probability that either of the two Porsches will win the race.

### Step 1: Interpret the given odds.
- For the first Porsche, the odds against winning are **3 to 1**.
  - This means the probability of winning for the first Porsche (\(P_1\)) is:
    \[
    P_1 = \frac{1}{3+1} = \frac{1}{4}
    \]
  
- For the second Porsche, the odds against winning are **5 to 3**.
  - This means the probability of winning for the second Porsche (\(P_2\)) is:
    \[
    P_2 = \frac{3}{5+3} = \frac{3}{8}
    \]

### Step 2: Calculate the probability that neither Porsche wins.
- The probability that the first Porsche does not win is:
  \[
  1 - P_1 = 1 - \frac{1}{4} = \frac{3}{4}
  \]

- The probability that the second Porsche does not win is:
  \[
  1 - P_2 = 1 - \frac{3}{8} = \frac{5}{8}
  \]

- The probability that neither Porsche wins (both fail to win) is:
  \[
  P(\text{Neither wins}) = \frac{3}{4} \times \frac{5}{8} = \frac{15}{32}
  \]

### Step 3: Calculate the probability that at least one Porsche wins.
- The probability that at least one Porsche wins is:
  \[
  P(\text{At least one wins}) = 1 - P(\text{Neither wins}) = 1 - \frac{15}{32} = \frac{17}{32}
  \]

### Step 4: Convert this probability into odds.
- The odds in favor of at least one Porsche winning are given by the ratio of the probability of the event happening to the probability of the event not happening:
  \[
  \text{Odds in favor of at least one Porsche winning} = \frac{P(\text{At least one wins})}{1 - P(\text{At least one wins})} = \frac{\frac{17}{32}}{\frac{15}{32}} = \frac{17}{15}
  \]

So, the odds in favor of either Porsche winning are **17 to 15**.

---

Would you like more details or have any other questions?

Here are some related questions to consider:

1. How are odds and probabilities related?
2. How do you calculate the probability of the complement of an event?
3. What are independent and dependent events in probability?
4. How do you calculate combined probabilities for independent events?
5. How do you convert probabilities into odds and vice versa?

**Tip:** Always remember that "odds against" is different from "odds in favor" when interpreting probabilities.

Learn how to calculate odds and probabilities for a road race involving two Porsches. Understand the concepts of odds against winning, probability calculations, and determining the odds for either Porsche winning. Suitable for high school students studying probability and odds.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation