ou purchase 5 tickets and enter into a raffle drawing that has 2 prizes. There are a total of87 tickets sold for the raffle.

No replacement: What is the probability of winning both prizes assuming you win the first prize and your ticket is not re-entered into the drawing?

With replacement: What is the probability of winning both prizes assuming you win the first prize and your ticket is re-entered into the drawing?What difference does it make?  Which gives the higher probability? 

### Without Replacement:
In this case, once you win the first prize, your winning ticket is **not re-entered** into the raffle. Here's how to calculate the probability of winning **both** prizes:

1. **Probability of winning the first prize:**  
   There are 87 tickets in total, and you own 5 of them. So, the probability of winning the first prize is:
   \[
   P(\text{First prize}) = \frac{5}{87}
   \]

2. **Probability of winning the second prize (after you win the first):**  
   Once you've won the first prize, there are 86 tickets left, but you now own only 4 remaining tickets. So, the probability of winning the second prize is:
   \[
   P(\text{Second prize | First prize}) = \frac{4}{86}
   \]

Thus, the total probability of winning **both prizes without replacement** is:
\[
P(\text{Both prizes without replacement}) = \frac{5}{87} \times \frac{4}{86} = \frac{20}{7482} \approx 0.002674
\]

### With Replacement:
Now, if your ticket **is re-entered** into the raffle after you win the first prize, the situation changes slightly. In this case:

1. **Probability of winning the first prize:**  
   This remains the same:
   \[
   P(\text{First prize}) = \frac{5}{87}
   \]

2. **Probability of winning the second prize (after you win the first):**  
   Since the ticket is re-entered, there are still 87 tickets in the raffle, and you still own 5 of them. So, the probability of winning the second prize is:
   \[
   P(\text{Second prize | First prize}) = \frac{5}{87}
   \]

Thus, the total probability of winning **both prizes with replacement** is:
\[
P(\text{Both prizes with replacement}) = \frac{5}{87} \times \frac{5}{87} = \frac{25}{7569} \approx 0.003303
\]

### Comparison:
- **Without replacement:** The probability is \( \approx 0.002674 \).
- **With replacement:** The probability is \( \approx 0.003303 \).

Clearly, the probability of winning both prizes is higher **with replacement** because you still have the same number of tickets in the second round.

### Key Difference:
- **Without replacement:** You lose a ticket after winning the first prize, reducing your chances for the second prize.
- **With replacement:** Your ticket is placed back into the pool, keeping your chances the same for both rounds.

Would you like further details or have any questions?

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Here are 5 related questions to deepen understanding:

1. What is the probability of winning exactly one prize without replacement?
2. How would the probability change if you had 10 tickets instead of 5?
3. What is the probability of winning no prizes at all in both cases?
4. How would the total number of tickets sold affect your chances?
5. What would happen to the probabilities if there were 3 prizes instead of 2?

**Tip:** Always clarify whether events are independent (like in the replacement case) or dependent (like in the no-replacement case) when calculating probabilities.

You purchase 5 tickets and enter into a raffle drawing that has 2 prizes. There are a total of 87 tickets sold for the raffle. No replacement: What is the probability of winning both prizes assuming you win the first prize and your ticket is not re-entered into the drawing? With replacement: What is the probability of winning both prizes assuming you win the first prize and your ticket is re-entered into the drawing? What difference does it make? Which gives the higher probability?

Calculate the probability of winning both prizes in a raffle with and without replacement. Explore key probability concepts, including dependent and independent events, using real-world raffle scenarios. Suitable for high school students studying probability.

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