To find the probability of getting between 40 and 60 heads when flipping a fair coin 100 times, we can use the normal approximation to the binomial distribution. Since the number of flips (n) is large, this approximation is quite effective.
1. **Understanding the Binomial Distribution**: Each flip of the coin has two possible outcomes: heads (H) or tails (T). The probability of getting heads (p) is 0.5, and similarly, the probability of tails is also 0.5. For 100 flips, we have:
- n (number of trials) = 100
- p (probability of success) = 0.5
- q (probability of failure) = 1 – p = 0.5
2. **Mean and Standard Deviation**: The mean (μ) and standard deviation (σ) of a binomial distribution can be calculated as follows:
- Mean (μ) = n * p = 100 * 0.5 = 50
- Standard Deviation (σ) = √(n * p * q) = √(100 * 0.5 * 0.5) = √25 = 5
3. **Normal Approximation**: According to the normal approximation, we can approximate the distribution of the number of heads when flipping the coin 100 times as a normal distribution with mean 50 and standard deviation 5. To find the probability of getting between 40 and 60 heads, we can standardize these values:
- Z-score for 40 heads = (40 – μ) / σ = (40 – 50) / 5 = -2
- Z-score for 60 heads = (60 – μ) / σ = (60 – 50) / 5 = 2
4. **Using Standard Normal Distribution**: To find the probability of getting between 40 and 60 heads, we can look up the Z-scores in the standard normal distribution table or use a calculator:
- Probability(Z < 2) ≈ 0.9772
- Probability(Z < -2) ≈ 0.0228
5. **Final Calculation**: Therefore, the probability of getting between 40 and 60 heads is:
Probability(40 < X < 60) = P(Z < 2) - P(Z < -2) = 0.9772 - 0.0228 = 0.9544
So, there is approximately a 95.44% chance of getting between 40 and 60 heads when flipping a coin 100 times. This probability illustrates the central limit theorem at work and emphasizes the tendency for results to cluster around the mean in large random trials.