Q.6.26 - A First Course in Probability

Question

Show that if n people are distributed at random along a road L miles long, then the probability that no 2 people are less than a distance D miles apart is when D … L/(n − 1), [1 − (n − 1)D/L] n. What if D > L/(n − 1)? 

Step-by-Step Solution

Answer

There will always be two person which will have their spaces crossed. The probability in this scenario is 0.

Step 1: Content Introduction

Use basic combinatoric argument to obtain the required probabilities.

Step 2: Content Explanation

Suppose that we can place first person uniformly wherever we want. Since, we want that everyone person has its private space of diameter D, we cannot place the first person in n - 1 balls of diameter D. So the probability that we will place him in some free space is simply

$1-\frac{(n-1)D}{L}$

Since, we have n persons, we have the probability that each of them has their own space. 

$\frac{(1-{(n-1)D)}^n}{L}$