Question

Each morning an individual leaves his house and goes for a run. He is equally likely to leave either from his front or back door. Upon leaving the house, he chooses a pair of running shoes (or goes running barefoot if there are no shoes at the door from which he departed). On his return he is equally likely to enter, and leave his running shoes, either by the front or back door. If he owns a total of \(k\) pairs of running shoes, what proportion of the time does he run barefooted?

Short answer

The individual never runs barefooted, as there must always be running shoes present at either the front door or the back door. The probability of running barefoot is 0.

Step-by-step solution

Represent the total number of running shoes and distribution of running shoes

Let k be the total number of running shoes. And let x be the number of running shoes present at the front door and y be the number of running shoes present at the back door. Therefore, \(k=x+y\) as all running shoes are distributed between the front and back doors.

Calculate the probability of selecting a door

Since the individual is equally likely to leave from the front or the back door, the probability of selecting a door can be denoted as: - Probability of choosing the front door: \(P(F) = \frac{1}{2}\) - Probability of choosing the back door: \(P(B) = \frac{1}{2}\)

Calculate the probabilities of finding no running shoes at the selected door

The probability of finding no running shoes at the front door with x running shoes is: \(P(\text{No Shoes} | F) = \frac{0}{x} = 0\) The probability of finding no running shoes at the back door with y running shoes is: \(P(\text{No Shoes} | B) = \frac{0}{y} = 0\)

Calculate the overall probability of running barefoot

We will now combine the probability of choosing a door with the probability of finding no running shoes at that chosen door. For this, we will use the total probability theorem: \(P(\text{No Shoes}) = P(\text{No Shoes} | F) \times P(F) + P(\text{No Shoes} | B) \times P(B)\) Plugging in the numbers, we get: \(P(\text{No Shoes}) = 0 \times \frac{1}{2} + 0 \times \frac{1}{2}\) \(P(\text{No Shoes}) = 0\) The answer is 0, meaning that he never runs barefooted, as there must always be running shoes present at either the front door or the back door.