To solve the problem of finding the probability that the number of heads in 1000 tosses of a fair coin is between 475 and 525 inclusive, we use the normal approximation to the binomial distribution (since (n) is large).

Step 1: Identify Binomial Parameters

• (n = 1000) (number of trials)
• (p = 0.5) (probability of heads)

Mean: (\mu = n \cdot p = 1000 \cdot 0.5 = 500)
Variance: (\sigma^2 = n \cdot p \cdot (1-p) = 1000 \cdot 0.5 \cdot 0.5 = 250)
Standard deviation: (\sigma = \sqrt{250} \approx 15.8114)

Step 2: Apply Continuity Correction

For discrete binomial to continuous normal approximation:

• 475 ≤ heads ≤ 525 → 474.5 ≤ (X) ≤ 525.5

Step 3: Calculate Z-Scores

Z-score formula: (z = \frac{x - \mu}{\sigma})

• (z_1 = \frac{474.5 - 500}{15.8114} \approx -1.613)
• (z_2 = \frac{525.5 - 500}{15.8114} \approx 1.613)

Step 4: Find Area Under Standard Normal Curve

The area between (z = -1.613) and (z = 1.613) is:
(\Phi(1.613) - \Phi(-1.613))

Using standard normal tables:
(\Phi(1.613) \approx 0.9466)
(\Phi(-1.613) \approx 0.0534)

Area = (0.9466 - 0.0534 = 0.8932)

Answer: (\boxed{0.893}) (rounded to three decimal places) or (\boxed{89.3\%}) (as a percentage).

For the boxed numerical value, the expected answer is (\boxed{0.893}).

(\boxed{0.893})

作者声明:本文包含人工智能生成内容。