Your friend challenges you to a game. You will flip a fair coin a specified number of times and if the proportion of heads is greater than $$ 40 \% $$, you will win the game. You get two choices-- flip the coin 25 times or flip the coin 250 times. Which of the two options should you choose to maximize your chances of winning and why? 250 times because as the number of flips increases, the proportion will get farther from the theoretical probability of 50\% 25 times because as the number of flips increases, the proportion will get closer to the theoretical probability of $$ 50 \% $$ 250 times because as the number of flips increases, the proportion will get closer to the theoretical probability of $$ 50 \% $$ 25 times because as the number of flips increases, the proportion will get farther from the theoretical probability of $$ 50 \% $$

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To determine which option maximizes your chances of winning the game, we need to analyze the behavior of the proportion of heads as the number of flips increases.

### Known Conditions:
1. You flip a fair coin (probability of heads = 0.5).
2. You win if the proportion of heads is greater than 40%.
3. You have two options: flip the coin 25 times or 250 times.

### Analysis:
1. **Law of Large Numbers**: As the number of trials (coin flips) increases, the sample proportion (the proportion of heads) will tend to get closer to the expected probability (which is 0.5 for a fair coin). This means that with more flips, the proportion of heads will stabilize around 50%.

2. **Standard Deviation of Proportion**: The standard deviation of the proportion of heads in a binomial distribution can be calculated using the formula:
   $$
   \sigma = \sqrt{\frac{p(1-p)}{n}}
   $$
   where $$ p $$ is the probability of heads (0.5) and $$ n $$ is the number of flips.

- For 25 flips:
     $$
     \sigma_{25} = \sqrt{\frac{0.5(1-0.5)}{25}} = \sqrt{\frac{0.25}{25}} = \sqrt{0.01} = 0.1
     $$

- For 250 flips:
     $$
     \sigma_{250} = \sqrt{\frac{0.5(1-0.5)}{250}} = \sqrt{\frac{0.25}{250}} = \sqrt{0.001} = 0.0316
     $$

3. **Proportion Greater than 40%**: 
   - For 25 flips, the proportion of heads can vary significantly due to the larger standard deviation (0.1). This means that there is a higher chance of the proportion being above 40% or below 40%.
   - For 250 flips, the smaller standard deviation (0.0316) means that the proportion of heads will be much closer to 50%, making it less likely to fall below 40%.

### Conclusion:
- **Choosing 250 flips** maximizes your chances of winning because the proportion of heads will be closer to the theoretical probability of 50%, and thus, it is less likely to fall below 40%.

Therefore, the correct choice is:
**250 times because as the number of flips increases, the proportion will get closer to the theoretical probability of $$ 50 \% $$.**
You should choose to flip the coin 250 times because as the number of flips increases, the proportion of heads gets closer to 50%, making it more likely to be greater than 40%.

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