Achieving 312 Heads

Solution & Explanation

Problem Breakdown

• Scenario: You flip a fair coin 576 times.
• Objective: Determine the probability of obtaining at least 312 heads.

This problem involves a binomial distribution, where:

• Number of trials (n): 576
• Probability of success (p) for each trial: 0.5 (since the coin is fair)

Binomial Distribution Characteristics

1. Mean (bc): $\mu = np = 576 \times 0.5 = 288$

2. Variance (c3^2): $\sigma^2 = np(1-p) = 576 \times 0.5 \times 0.5 = 144$

3. Standard Deviation (c3): $\sigma = \sqrt{144} = 12$

Normal Approximation

Given the large number of trials (n = 576), the binomial distribution can be approximated using a normal distribution due to the Central Limit Theorem.

• Standard Normal Distribution:
  • Mean (bc): 288
  • Standard Deviation (c3): 12

Standardization

To find the probability of getting at least 312 heads, we convert the problem into a standard normal distribution problem by calculating the Z-score:

• Z-score for 312 heads: $Z = \frac{312 - 288}{12} = \frac{24}{12} = 2$

Probability Calculation

In a standard normal distribution, the probability that a value is greater than two standard deviations above the mean (Z = 2) is approximately 2.5%.

• Interpretation:
  • The area to the right of Z = 2 corresponds to the probability of obtaining at least 312 heads.
  • Therefore, the probability of obtaining at least 312 heads in 576 flips is approximately 2.5%.

Conclusion

By using the normal approximation to the binomial distribution, we find that the probability of flipping at least 312 heads in 576 tosses of a fair coin is about 2.5%. This solution is based on the properties of the normal distribution, where approximately 95% of data lies within two standard deviations from the mean, leaving 5% in the tails, and thus 2.5% in the right tail.