生日问题变体：600人中至少75人九月出生的近似概率求解

Great question—you’re spot-on to model this with the binomial distribution for approximations, given the large sample size. Let’s break down both parts (a and b) using normal approximation to the binomial, which works well here because our sample size n is large enough that np and n(1-p) are both significantly greater than 5.

Part a: Equal Probability per Month #

First, if each month has an equal probability of being a birth month:

• Total trials: n = 600
• Probability of "success" (birth in September): p = 1/12 ≈ 0.0833

For the binomial distribution, we calculate the mean and variance first:

• Mean: μ = np = 600 * (1/12) = 50
• Variance: σ² = np(1-p) = 50 * (11/12) ≈ 45.833
• Standard deviation: σ ≈ √45.833 ≈ 6.77

Since we’re approximating a discrete binomial variable with a continuous normal distribution, we need to apply continuity correction. We want P(X ≥ 75) (discrete), which translates to P(Y ≥ 74.5) where Y ~ N(μ, σ²).

Calculate the z-score:

z = (74.5 - μ) / σ ≈ (74.5 - 50) / 6.77 ≈ 3.62

Looking up this z-score in the standard normal table:

• Φ(3.62) ≈ 0.99984 (the cumulative probability up to z=3.62)
• So P(Y ≥ 74.5) = 1 - Φ(3.62) ≈ 1 - 0.99984 = 0.00016

That’s an approximate probability of 0.016%.

Part b: Equal Probability per Day #

Now, if each day has equal probability (we’ll use 365 days for simplicity, ignoring leap years for approximation):

• September has 30 days, so p = 30/365 ≈ 0.08219
• Total trials still n = 600

Calculate mean and variance:

• Mean: μ = np = 600 * (30/365) ≈ 49.315
• Variance: σ² = np(1-p) ≈ 49.315 * (335/365) ≈ 45.26
• Standard deviation: σ ≈ √45.26 ≈ 6.73

Again, apply continuity correction for P(X ≥ 75) ≈ P(Y ≥ 74.5):

z = (74.5 - 49.315) / 6.73 ≈ 25.185 / 6.73 ≈ 3.74

Looking up z=3.74:

• Φ(3.74) ≈ 0.99991
• So P(Y ≥ 74.5) = 1 - 0.99991 = 0.00009

This gives an approximate probability of 0.009%.

A quick note: Both probabilities are extremely low because 75 is far above the expected number of September births (around 50 in both cases), which makes intuitive sense—we’re looking at a tail event way out in the distribution.

内容的提问来源于stack exchange，提问作者doddy