Answer to Question #159435 in Statistics and Probability for Furne

Question

A parking lot has 10 parking spaces arranged in a row. There are 7 cars parked. Assume that each car owner has picked at a random a parking place among the spaces available. Specify an appropriate sample space and determine the probability that the three empty places are adjacent to each other.

Answer 1

There are 10 parking spaces arranged in a row. There are 7 cars parked. Take an unordered sample space whose outcomes are given by all possible sets of the three different parking places out of the ten parking places.

The number of outcomes of the sample space is given by

$n(S) = \frac{10!}{3!7!}= \frac{10 \times 9 \times 8}{3 \times 2} = 10 \times 3 \times 4 = 120$

Each outcome of the sample space gets assigned the same probability of $\frac{1}{120}$

The number of outcomes with three adjacent parking places is 8.

The desired probability is

$P = \frac{8}{120}=\frac{1}{15}=0.0666$

Answer: 0.0666