Question 1180015: Suppose 200 people are lined up side-by-side, each one holding a fair coin. Each person flips their coin 64 times; every time it lands heads they step 1 meter forward, each time it lands tails they step 1 meter backward. Use a normal approximation to answer the following question: after everyone finishes their 64 steps, approximately how many people will be standing between 4 and 8 meters behind the starting line?
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
Answer by ikleyn(52805)  (Show Source):
You can put this solution on YOUR website! .
I agree with the solution by @CPhill almost everywhere, except of its final part.
After the number of interest of heads is determined to be between 28 and 30 inclusive,
the continuity correction should be applied.
Thus, actually, we look for the probability for H to be between 27.5 and 30.5.
So, in the solution by @CPhill the z-scores z1 and z2 should be modified consistently.
The final answer for the probability is then 0.2235 instead of 0.1498,
and the number of people between -8 and -4 meters is then about 0.2235*200 = 44.7,
or about 45.
This difference is significant and, therefore, should be accounted.
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