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'll use the odds given and convert them to probabilities. Then, we'll determine the combined probability that either car will win.

### Step 1: Convert odds to probabilities
1. **First car's odds:** The odds against the first car winning are 3 to 1.
   - Probability of the first car winning, \( P_1 \), is given by:
     \[
     P_1 = \frac{1}{3+1} = \frac{1}{4}
     \]
   - So, \( P_1 = 0.25 \).

2. **Second car's odds:** The odds against the second car winning are 5 to 3.
   - Probability of the second car winning, \( P_2 \), is given by:
     \[
     P_2 = \frac{3}{5+3} = \frac{3}{8}
     \]
   - So, \( P_2 = 0.375 \).

### Step 2: Calculate the probability that either car will win
To find the probability that either car will win (denoted as \( P(\text{either}) \)), we'll use the formula:
\[
P(\text{either}) = P_1 + P_2 - P_1 \times P_2
\]
Substituting the values we found:
\[
P(\text{either}) = 0.25 + 0.375 - (0.25 \times 0.375)
\]
\[
P(\text{either}) = 0.625 - 0.09375 = 0.53125
\]

### Step 3: Convert the probability back to odds
The probability \( P(\text{either}) = 0.53125 \) corresponds to odds in favor of either car winning as:
\[
\text{Odds in favor} = \frac{P(\text{either})}{1 - P(\text{either})} = \frac{0.53125}{1 - 0.53125} = \frac{0.53125}{0.46875} \approx 1.133
\]
Thus, the odds can be approximated as \( \text{odds in favor} \approx 1.133 \text{ to } 1 \).

Since we typically express odds in whole numbers, the odds that either car will win are approximately **1.133 to 1** or, for consistency, can be reported as **113 to 100**.

Would you like further details or have any questions?

Here are five related questions:
1. How would the result change if the odds against one of the cars were different?
2. What if the odds were given in decimal format instead of ratios?
3. How would we calculate the odds if more than two cars were involved?
4. What is the difference between probability and odds?
5. How can these concepts be applied to other scenarios like gambling or risk assessment?

**Tip:** Understanding the relationship between probability and odds is crucial for interpreting risk and likelihood in various contexts.

Learn how to calculate probabilities and odds for a problem involving two Porsches in a road race in Italy. Understand the conversion between odds and probabilities, and apply these concepts to determine the odds of either car winning.

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