Question

1. A robbery has just been committed at the Corner Market in the downtown area of the city. The market owner was able to activate the alarm, and the robber fled on foot. Police officers arrived a few minutes later and asked the owner, “How long ago did the robber leave?” “He left about 9 minutes ago and headed east.”, the owner responded. “He’s probably nine blocks away by now.”, one of the officers said to the other. “Not likely”, said the store owner. “He was so stoned on drugs that I bet he’s still only within a few blocks of here! He’s probably just running in circles!”

Perform a simulation experiment that will test the store owner’s hypothesis. Assume the following:

• It takes him 1 minute to walk each block (they are short).
• At the end of each minute interval he is at a corner. There is a 32% chance that he will fall asleep for 1 minute, and an 17% chance that he will walk east, or north, or west, or south. 3

My suggestion is to run at least 1000 trials of the nine-minute journey of the robber.

1. On average how many block has he walked?
2. On average how far away is he (using blocks - not “diagonals”) from the store?
3. What is the probability he is two blocks or less (walking on streets) from the store?
4. What is a 95% confidence interval estimate of this probability?

Answer 1

Running the simulations, we get:

a. Average blocks walked = 6.126

Note that if he walks back a block, it is still counted. So, this is the steps he is not sleeping

b. Average distance from the store = 2.744

Note that this is the block distance from the store.

c. Probability that he is two blocks or less from the store = 0.4670

0d. The standard deviation of the probability s is 0.4989

The sample size n is 1000. Since the number of samples is large, we use the normal distribution.

A 95% confidence interval CI corresponds to a z-score of 1.96. So, the CI will be:

Note: Results will vary slightly if the simulation is done again.

Here is a python program for parts a to c:

from random import random
import numpy as np

cnt = 1000
smbwk = 0
smwlk = 0
awlk = []

for simulation in range(cnt):
    x = 0
    y = 0
    bwk = 0

    for i in range(9):
        p = random()
        if p <= 0.32:
            pass
        elif p <= 0.49:
            x += 1
            bwk += 1
        elif p <= 0.66:
            y += 1
            bwk += 1
        elif p <= 0.83:
            x -= 1
            bwk += 1
        elif p <= 1.00:
            y -= 1
            bwk += 1

    wlk = abs(x) + abs(y)

    smbwk += bwk
    smwlk += wlk
    if wlk <= 2:
        pwlk = 1
    else:
        pwlk = 0
    awlk.append(pwlk)

print(smbwk/cnt)
print(smwlk/cnt)
print(np.mean(awlk))
print(np.std(awlk))