If 3 winning tickets are drawn from an urn of 100 tickets, then what is the probability of winning a person who buys 4 tickets?If 3 winning tickets are drawn from an urn of 100 tickets, then what is the probability of winning a person who buys 4 tickets?

To solve the problem of finding the probability that a person who buys 4 tickets wins at least one of the 3 winning tickets drawn from an urn of 100 tickets, follow these steps:

### Step 1: Identify the total number of outcomes
- The total number of ways to choose 3 winning tickets from 100 tickets is given by the combination formula \( \binom{100}{3} \):

\[
\binom{100}{3} = \frac{100 \times 99 \times 98}{3 \times 2 \times 1} = 161700
\]

### Step 2: Calculate the number of favorable outcomes
- The number of favorable outcomes is the number of ways to get at least one winning ticket when buying 4 tickets.
- To find this, it is easier to first calculate the number of ways in which none of the 4 tickets are winning tickets, and then subtract this from the total possible outcomes.

#### Step 2a: Calculate the number of ways to choose 3 winning tickets from the 96 non-winning tickets
- If none of the 4 tickets are winning tickets, then all 3 winning tickets must be among the remaining 96 tickets. The number of ways to choose 3 winning tickets from 96 tickets is given by \( \binom{96}{3} \):

\[
\binom{96}{3} = \frac{96 \times 95 \times 94}{3 \times 2 \times 1} = 142880
\]

#### Step 2b: Calculate the number of ways to win at least one ticket
- The number of ways to win at least one ticket is the complement of the number of ways to win zero tickets. Therefore:

\[
\text{Favorable outcomes} = \text{Total outcomes} - \text{Unfavorable outcomes} = 161700 - 142880 = 18820
\]

### Step 3: Calculate the probability of winning
- The probability of winning at least one ticket is the ratio of the favorable outcomes to the total outcomes:

\[
\text{Probability} = \frac{18820}{161700} \approx 0.1164
\]

So, the probability that a person who buys 4 tickets wins at least one of the 3 winning tickets is approximately \( 0.1164 \), or about \( 11.64\% \).

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Would you like more details or have any questions? Here are 5 related questions:

1. How do you calculate the probability of winning if the number of tickets purchased changes?
2. What is the complement rule in probability, and how is it applied?
3. Can you explain why it’s easier to calculate the probability of losing first in some cases?
4. What is the probability of winning exactly one ticket?
5. How would the problem change if more or fewer winning tickets were drawn from the urn?

**Tip:** When dealing with probabilities involving "at least one" outcomes, it's often easier to first calculate the complement (the probability of the opposite event) and then subtract from 1.

Explore how to calculate the probability of winning when buying multiple tickets against a set number of winning tickets drawn from an urn. Understand the combination formula and step-by-step calculation for finding the probability. Suitable for Grades 10-12.

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