A lottery consists of selecting 5 numbers out of 30 numbers. You win $10 if exactly three of your 5 numbers are matched to the winning numbers chosen. What is the probability of winning the $10?

Round your answer to six decimal places.



To find the probability of winning $10 in this lottery, we need to calculate the probability of exactly 3 out of your 5 chosen numbers matching the 5 winning numbers.

Here’s the step-by-step breakdown:

1. **Total Number of Ways to Choose Winning Numbers:**

   The total number of ways to choose 5 numbers out of 30 is given by the combination formula:
   \[
   \text{Total ways} = \binom{30}{5}
   \]

2. **Number of Favorable Outcomes:**

   To win $10, exactly 3 of your 5 chosen numbers must match the winning numbers, and the remaining 2 numbers must not match. We calculate the number of favorable outcomes as follows:

   - The number of ways to choose 3 matching numbers from the 5 winning numbers:
     \[
     \text{Ways to choose 3 matches} = \binom{5}{3}
     \]
   
   - The number of ways to choose 2 non-matching numbers from the remaining 25 numbers (since 5 numbers are already chosen as winning numbers):
     \[
     \text{Ways to choose 2 non-matches} = \binom{25}{2}
     \]

   Therefore, the total number of favorable outcomes is:
   \[
   \text{Favorable outcomes} = \binom{5}{3} \times \binom{25}{2}
   \]

3. **Probability of Winning $10:**

   The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes:
   \[
   \text{Probability} = \frac{\binom{5}{3} \times \binom{25}{2}}{\binom{30}{5}}
   \]

Let’s calculate these values.

First, calculate the combinations:

- $\binom{30}{5} = \frac{30!}{5!(30-5)!}$
- $\binom{5}{3} = \frac{5!}{3!(5-3)!}$
- $\binom{25}{2} = \frac{25!}{2!(25-2)!}$

Then, we can compute the probability. I'll do the math now.The probability of winning $10 in the lottery by matching exactly three of your five chosen numbers is approximately **0.021052**.

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. How would the probability change if the lottery involved selecting 6 numbers instead of 5?
2. What is the probability of matching all 5 numbers in this lottery?
3. How would the probability be affected if there were 40 numbers to choose from instead of 30?
4. What is the expected value of this lottery game?
5. How does this probability compare to a standard 6/49 lottery game?

**Tip:** Understanding combination formulas is essential for solving probability problems involving selections, such as lotteries.

A lottery consists of selecting 5 numbers out of 30 numbers. You win $10 if exactly three of your 5 numbers are matched to the winning numbers chosen. What is the probability of winning the $10?

Learn how to calculate the probability of winning $10 in a lottery where you select 5 numbers out of 30, and win if exactly 3 of your numbers match the winning ones. This detailed solution covers key concepts in probability and combinatorics, using the combination formula and probability ratio. Suitable for Grades 10-12.

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