Value of Coin Flip Product

Solution & Explanation

To solve the problem of calculating the expected value of the product of the number of heads and tails when a coin is flipped 100 times, we need to delve into the properties of the binomial distribution. Let's break it down step-by-step:

Understanding the Problem

• Coin Tosses: We have 100 independent coin tosses.
• Heads and Tails: Let $H$ be the number of heads and $T = 100 - H$ be the number of tails.
• Objective: Calculate $E[H \times T]$, the expected value of the product of heads and tails.

Binomial Distribution Basics

• Binomial Distribution: For a fair coin, $H \sim \text{Binomial}(n=100, p=0.5)$.
• Expected Value: $E[H] = np = 100 \times 0.5 = 50$.
• Variance: $\text{Var}(H) = np(1-p) = 100 \times 0.5 \times 0.5 = 25$.

Calculating $E[H \times T]$

1. Expression: $E[H \times T] = E[H \times (100 - H)] = E[100H - H^2]$.
2. Using Linearity of Expectation:
   • $E[100H - H^2] = 100E[H] - E[H^2]$.
3. Finding $E[H^2]$:
   • $\text{Var}(H) = E[H^2] - (E[H])^2$.
   • $25 = E[H^2] - 50^2$.
   • $E[H^2] = 25 + 2500 = 2525$.
4. Substitute Back:
   • $E[H \times T] = 100 \times 50 - 2525 = 2475$.

Confidence Interval for $E[H \times T]$

• Central Limit Theorem: For large $n$, the distribution of $H$ is approximately normal.
• Standard Error: $\text{SE} = \sqrt{\text{Var}(H \times T)/n}$.
• Approximate $\text{Var}(H \times T)$:
  • Simplification using $H$ and $T$ as approximately normal gives $\text{Var}(H \times T) \approx 100^2 \times 0.5 \times 0.5 \times (1 - 0.5) = 2475$.
• Confidence Interval Calculation:
  • $95\% \text{ CI} = [\text{mean} - 1.96 \times \text{SE}, \text{mean} + 1.96 \times \text{SE}]$.
  • $\text{SE} \approx \sqrt{\frac{2475}{100}} \approx 4.975$.
  • $95\% \text{ CI} = [2475 - 1.96 \times 4.975, 2475 + 1.96 \times 4.975] \approx [2465.25, 2484.75]$.

Conclusion

The expected value of the product of the number of heads and tails in 100 coin flips is 2475, with a 95% confidence interval approximately between 2465.25 and 2484.75. This reflects the variability and normal approximation of the binomial distribution at this scale.