SOLUTION: A car is parked among N cars in a row, not at either end. On this return the owner finds that exactly r of the N places are still occupied. What is the probability that both neigh

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Question 1177980: A car is parked among N cars in a row, not at either end. On this return the owner finds that exactly r
of the N places are still occupied. What is the probability that both neighbouring places are empty.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2189) (Show Source):
You can put this solution on YOUR website!
Let's break down this problem step-by-step.
Understanding the Problem
Total Cars: N
Our Car: Parked among N cars, not at either end.
Remaining Cars: r (including our car)
Goal: Find the probability that both neighboring places are empty.
1. Total Possible Arrangements
We know that r spots are occupied, including our car. Since our car's position is fixed, we need to choose the remaining r-1 spots from the N-1 available spots (excluding our car).
Total ways to choose r-1 spots from N-1 spots: C(N-1, r-1) = (N-1)! / [(r-1)! * (N-r)!]
2. Favorable Arrangements
We want to find the arrangements where our car's two neighboring spots are empty.
Fix Our Car: Our car's position is fixed.
Empty Neighbors: The two spots next to our car must be empty.
Remaining Spots: We have N - 3 remaining spots (excluding our car and its neighbors).
Remaining Cars: We need to choose r - 1 spots from the N - 3 spots.
Favorable ways to choose r-1 spots from N-3 spots: C(N-3, r-1) = (N-3)! / [(r-1)! * (N-r-2)!]
3. Calculate the Probability
The probability is the ratio of favorable arrangements to total arrangements:
Probability = C(N-3, r-1) / C(N-1, r-1)
Probability = [(N-3)! / ((r-1)! * (N-r-2)!)] / [(N-1)! / ((r-1)! * (N-r)!)]
Probability = [(N-3)! / (N-r-2)!] * [(N-r)! / (N-1)!]
Probability = [(N-3)! / (N-1)!] * [(N-r)! / (N-r-2)!]
Probability = [1 / ((N-1)(N-2))] * [(N-r)(N-r-1)]
Probability = [(N-r)(N-r-1)] / [(N-1)(N-2)]
Therefore, the probability that both neighboring places are empty is [(N-r)(N-r-1)] / [(N-1)(N-2)].

Answer by ikleyn(53763) (Show Source):
You can put this solution on YOUR website!
.
A car is parked among N cars in a row, not at either end. On this return the owner finds that exactly r
of the N places are still occupied. What is the probability that both neighboring places are empty.
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My solution here is another way to organize your thoughts and to present a solution.

For clarity, I will re-formulate the problem in this way.

In a parking lot, there are N parking places, in a row. Originally, they all are occupied by cars. The owner's car is parked among these N cars, not at either end. On his return the owner finds that exactly r of the N places are still occupied. What is the probability that both neighboring places are empty ?

Below is the solution for this modified formulation.

There are N places in a row at the table. One place is marked "C" (symbolizing your car), which is not at either end. You have (N-r) cards in your hands with letter E (symbolizing "empty place"). You distribute these r cards randomly over all the unmarked places. What is the probability that two neighboring places to "C" will be "E". The probability that the left place is card "E" is . The probability that the right place is card "E" is then . The overall probability that both neighbouting cards are "E", is the product P = . ANSWER

Solved.