Question

A parking lot consists of a single row containing n parking spaces (n ≥ 2). Mary arrives when all spaces are free. Tom is the next person to arrive. Each person makes an equally likely choice among all available spaces at the time of arrival. Describe the sample space. Obtain P(A), the probability the parking spaces selected by Mary and Tom are at most 2 spaces apart.

Answer 1

a) The total sample space of possible parking = n (n-1)

b)pr(at most 2 spaces) = 6/n

Explanation:

a) Since Mary was the first person to arrive at the parking lot, she has n available parking spaces.

On arrival, Tom would meet a parking space that is already occupied by Mary, so he has (n-1) available parking spaces

The total sample space of possible parking = n (n-1)

b) Probability that the parking spaces selected by Mary and Tom are at most 2 spaces apart

Pr(at most 2) = pr(0 space) + pr(1 space) + pr(2 spaces)

The number of possible placings for 0 space = 2(n-0) = 2n

The factor 2 caters for the reversal in the order in which the vehicles are placed.

pr(0 space) = 2n/n(n-1) = 2/(n-1)

Number of possible placings for 1 space = 2(n-1)

pr(1 space) = 2(n-1)/n(n-1) = 2/n

The number of possible placings for 2 spaces = 2(n-2)

pr(2 spaces) = 2(n-2)/n(n-1)

pr(at most 2 spaces) = 2/n-1 + 2/n + 2(n-2)/n(n-1)

pr(at most 2 spaces) = [ 2n + 2(n-1) + 2(n-2)]/[n(n-1)]

pr(at most 2 spaces) = 6/n