Question 1177980
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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.


 <pre>
    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 ?
</pre>


Below is the solution for this modified formulation.



<pre>
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  {{{(N-r)/(N-1)}}}.


The probability that the right place is card "E" is then  {{{(N-r-1)/(N-2)}}}.


The overall probability that both neighbouting cards are "E", is the product  

    P = {{{((N-r)/(N-1))*((N-r-1)/(N-2))}}}.    <U>ANSWER</U>
</pre>

Solved.