You flip a fair coin 10,000 times. Approximate the probability that
the di�fference between the number of heads and number of tails is at most 100.
To approximate the probability that the difference between the number of heads and tails is at most 100 when flipping a fair coin 10,000 times, we can use a normal approximation to the binomial distribution.
The first step is to calculate the mean and standard deviation of the binomial distribution. In this case, the mean (μ) is the product of the number of trials (n) and the probability of success (p). Since we have a fair coin, the probability of heads (p) is 0.5, and the number of trials (n) is 10,000.
μ = n * p = 10,000 * 0.5 = 5,000
The standard deviation (σ) of the binomial distribution is the square root of the product of the number of trials, the success probability, and the failure probability (q = 1 - p).
σ = sqrt(n * p * q) = sqrt(10,000 * 0.5 * 0.5) = sqrt(2,500) = 50
Next, we use the normal approximation to estimate the probability. We convert the original problem to a standard normal distribution by subtracting the mean and dividing by the standard deviation.
p(X ≤ 100) = p((X - μ) / σ ≤ (100 - 5,000) / 50) = p(Z ≤ -98)
Now, we can look up the area under the standard normal distribution curve for Z ≤ -98 using a z-table or calculator. However, the value of -98 is very extreme and falls well beyond the typical range of the normal distribution. As a result, the probability is expected to be extremely small and close to 0.
Therefore, the approximate probability that the difference between the number of heads and tails is at most 100 when flipping a fair coin 10,000 times is essentially 0.