Question 1180015
Here's how to solve this problem using a normal approximation:

**1. Define the Random Variable:**

* Let X be the number of heads in 64 coin flips.
* X follows a binomial distribution with n = 64 and p = 0.5.
* Each person's final position is determined by the difference between the number of heads and tails.

**2. Relate Position to Heads and Tails:**

* Let H be the number of heads and T be the number of tails.
* Final position = H - T
* We know H + T = 64, so T = 64 - H
* Final position = H - (64 - H) = 2H - 64

**3. Determine the Range of Heads:**

* We want to find the number of people between -8 and -4 meters.
* -8 ≤ 2H - 64 ≤ -4
* 56 ≤ 2H ≤ 60
* 28 ≤ H ≤ 30

**4. Approximate with a Normal Distribution:**

* The binomial distribution can be approximated by a normal distribution when n is large enough.
* Mean (μ) = np = 64 * 0.5 = 32
* Standard deviation (σ) = √(np(1-p)) = √(64 * 0.5 * 0.5) = √16 = 4

**5. Calculate Z-scores:**

* For H = 28: z1 = (28 - 32) / 4 = -1
* For H = 30: z2 = (30 - 32) / 4 = -0.5

**6. Find the Probability:**

* Use a standard normal distribution table or calculator to find the area between z1 and z2.
* P(-1 ≤ Z ≤ -0.5) = P(Z ≤ -0.5) - P(Z ≤ -1)
* P(Z ≤ -0.5) ≈ 0.3085
* P(Z ≤ -1) ≈ 0.1587
* P(-1 ≤ Z ≤ -0.5) ≈ 0.3085 - 0.1587 = 0.1498

**7. Calculate the Number of People:**

* Multiply the probability by the total number of people: 200 * 0.1498 ≈ 29.96

**Answer:**

Approximately 30 people will be standing between 4 and 8 meters behind the starting line.