To determine the probability of getting between 40 and 60 heads when flipping a fair coin 100 times, we need to use the normal approximation to the binomial distribution.
When we flip a coin, the outcome is binary: heads or tails. Each flip has a probability of 0.5 for heads. For a binomial distribution, the mean (µ) and standard deviation (σ) can be calculated using the formulas:
- Mean (µ) = n * p = 100 * 0.5 = 50
- Standard deviation (σ) = √(n * p * (1 – p)) = √(100 * 0.5 * 0.5) = √25 = 5
Next, we need to find the z-scores for 40 and 60 heads:
- For 40 heads: Z = (40 – 50) / 5 = -2
- For 60 heads: Z = (60 – 50) / 5 = 2
Now, we look up the z-scores in the standard normal distribution table:
- P(Z < -2) ≈ 0.0228
- P(Z < 2) ≈ 0.9772
To find the probability of getting between 40 and 60 heads, we calculate:
P(40 < X < 60) = P(Z < 2) - P(Z < -2) = 0.9772 - 0.0228 = 0.9544
This means that there is approximately a 95.44% chance of getting between 40 and 60 heads when you flip a coin 100 times. It’s important to remember that this method uses a normal approximation, which is generally accurate for a large number of trials, like 100 flips in this case.