Twenty tickets are sold in a raffle which has 2 prizes. If you buy 5 tickets, the probability of winning at least one prize is:

To determine the probability of winning at least one prize in a raffle where 20 tickets are sold and 2 prizes are available, and you purchase 5 of those tickets, we can use the complementary probability method.

Let's denote the following:

- \( n = 20 \) (total number of tickets)
- \( k = 2 \) (number of prizes)
- \( m = 5 \) (number of tickets you buy)

We aim to find the probability of winning at least one prize. It is easier to first find the probability of not winning any prize and then subtract this from 1.

### Step-by-Step Solution

1. **Calculate the total number of ways to choose 2 prizes out of 20 tickets:**
   \[
   \binom{20}{2} = \frac{20!}{2!(20-2)!} = \frac{20 \times 19}{2 \times 1} = 190
   \]

2. **Calculate the number of ways to choose 2 prize-winning tickets from the remaining 15 tickets (if none of your 5 tickets win):**
   \[
   \binom{15}{2} = \frac{15!}{2!(15-2)!} = \frac{15 \times 14}{2 \times 1} = 105
   \]

3. **Calculate the probability of not winning any prize:**
   \[
   P(\text{no prize}) = \frac{\binom{15}{2}}{\binom{20}{2}} = \frac{105}{190} = \frac{21}{38}
   \]

4. **Calculate the probability of winning at least one prize:**
   \[
   P(\text{at least one prize}) = 1 - P(\text{no prize}) = 1 - \frac{21}{38} = \frac{17}{38}
   \]

Thus, the probability of winning at least one prize when you buy 5 tickets out of 20 in a raffle with 2 prizes is:
\[
\boxed{\frac{17}{38}}
\]

Would you like further details or have any questions?

### Relative Questions:
1. What is the probability of winning both prizes with 5 tickets out of 20?
2. How would the probability change if you bought 6 tickets instead of 5?
3. Can we generalize this method to a raffle with \( n \) tickets and \( k \) prizes?
4. How do combinatorial coefficients \(\binom{n}{k}\) work?
5. What is the probability of not winning any prize with a single ticket?
6. How does the probability change with more prizes but the same number of tickets?
7. What if the raffle has a larger number of tickets, like 100 with 10 prizes, and you bought 10 tickets?
8. How would you calculate the expected value of the number of prizes won?

### Tip:
When calculating probabilities involving combinations, remember that the complementary probability method (finding the probability of the complementary event and subtracting from 1) can simplify many problems.

Explore the probability of winning at least one prize in a raffle with 20 tickets and 2 prizes when buying 5 tickets. Learn about combinatorial coefficients and the complementary probability method used to solve the problem. Suitable for high school students studying probability and combinatorics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation