7. Hawkeye Elementary School is having a fundraising raffle. The school leaders are voting to choose between the two options below:

Option 1: Raffle off two prizes and include all boys and girls in both raffles.
Option 2: Have a separate raffle for boys and a separate raffle for girls with 1 prize each.

\begin{tabular}{|l|r|}
\hline
Girls & 171 \\
\hline
Boys & 147 \\
\hline
Total & 318 \\
\hline
\end{tabular}

Suppose you get to vote and you have a daughter in sports. Which option would you vote for and why? Use at least two calculations to justify your answer.



Answer :

Let's analyze both options step-by-step to determine which option would be more favorable for your daughter to win a prize.

### Option 1: Combined Raffle with Two Prizes
In this option, we raffle off two prizes among all students (boys and girls), with the probability of winning calculated as follows:

1. Probability of a girl winning the first prize:
- The number of girls is 171 out of a total of 318 students.
- Probability (P) is given by:
[tex]\[ P(\text{girl wins first prize}) = \frac{171}{318} \approx 0.5377 \][/tex]

2. Probability of a girl winning the second prize:
- Case 1: If a girl wins the first prize.
[tex]\[ P(\text{girl wins second prize after girl}) = \frac{170}{317} \approx 0.5363 \][/tex]
- Case 2: If a boy wins the first prize.
[tex]\[ P(\text{girl wins second prize after boy}) = \frac{171}{317} \approx 0.5394 \][/tex]

3. Total probability for at least one girl winning a prize (combined raffle):
- This combines the probabilities of either a girl winning the first prize or a girl winning the second prize after a boy wins the first prize.
[tex]\[ P(\text{girl wins at least one prize}) = P(\text{girl wins first prize}) + (1 - P(\text{girl wins first prize})) \times P(\text{girl wins second prize after boy}) \][/tex]
[tex]\[ P(\text{girl wins at least one prize}) \approx 0.5377 + (1 - 0.5377) \times 0.5394 \approx 0.7871 \][/tex]

### Option 2: Separate Raffles for Boys and Girls
In this option, there will be a separate raffle for boys and a separate raffle for girls with 1 prize each. The probability of a girl winning the prize is:

4. Probability of a girl winning the prize when raffled separately:
- Since the raffle for girls includes only female students, and there is one prize available, the probability is:
[tex]\[ P(\text{girl wins separate raffle}) = 1.0 \][/tex]

### Conclusion
Comparing the probabilities:

- Option 1 (Combined raffle): The probability of a girl winning at least one prize is approximately \(0.7871\) or 78.71%.
- Option 2 (Separate raffles): The probability of a girl winning the prize in the separate raffle is \(1.0\) or 100%.

Given that you have a daughter in sports and you want to maximize her chances of winning a prize, Option 2 is clearly the better choice. The separate raffles guarantee that a girl will win the prize in the girl's raffle, ensuring a 100% chance of a girl (which could be your daughter) winning. This option provides the highest probability of success for your daughter.

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