When n is sufficiently large, the binomial distribution B(n, p) approximately follows the normal distribution N(np, np(1−p)) (the central limit theorem).
Example ① Estimate the probability of getting heads at least 60 times in 100 coin tosses.
Solution:B(100, 0.5) ≒ N(50, 25). σ=5.
Z = (60−50)/5 = 2. P(Z≧2) ≒ 0.025 (2.5%)
Solution:B(100, 0.5) ≒ N(50, 25). σ=5.
Z = (60−50)/5 = 2. P(Z≧2) ≒ 0.025 (2.5%)
Key Point
- As a guideline, np ≧ 5 and n(1−p) ≧ 5
Practice Problems
- Find the standard deviation of B(400, 0.5).
Solutions and Explanations
- Solution:10
√(400·0.25)=10.