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.
