Mary and Tom park their cars in an empty parking lot with n>=2 consecutive parking spaces (i.e, n spaces in a row, where only one car fits in each space). Mary and Tom pick parking spaces at random. (All pairs of parking spaces are equally likely.) What is the probability that there is at most one empty parking space between them?
To find the probability that there is at most one empty parking space between Mary and Tom's cars, we need to consider two scenarios: one where there is no empty space between them, and one where there is exactly one empty space between them.
Let's assume that the empty parking lot has n consecutive parking spaces. Each parking space can be labeled as either occupied (O) or empty (E). We can represent the parking lot arrangement using a string of letters where each letter represents the state of a parking space (O or E).
For the first scenario, where there is no empty space between Mary and Tom's cars, we can analyze the possible arrangements. Since Mary and Tom pick parking spaces at random, they can place their cars at any two consecutive spaces. The remaining n-2 spaces can either be occupied or empty.
The number of arrangements for the first scenario is given by:
Arrangements_scenario1 = 2 (occupied parking spaces for Mary and Tom) * 1 (only one possible arrangement for the remaining parking spaces) = 2
For the second scenario, where there is exactly one empty space between Mary and Tom's cars, we need to analyze the possible arrangements. Again, Mary and Tom can place their cars at any two consecutive spaces, but now there needs to be exactly one empty space between them. The remaining n-3 spaces can either be occupied or empty.
The number of arrangements for the second scenario is given by:
Arrangements_scenario2 = 2 (occupied parking spaces for Mary and Tom) * (n-2) (number of ways to arrange the remaining occupied and empty parking spaces) = 2(n-2)
Now, to find the total number of possible arrangements for the parking lot, we need to consider all possible arrangements of n parking spaces, regardless of whether there are empty spaces between Mary and Tom's cars.
Total_number_of_arrangements = 2^n (since each parking space can be either occupied or empty)
Finally, to calculate the probability, we can add the number of arrangements for the two scenarios and divide it by the total number of arrangements:
Probability = (Arrangements_scenario1 + Arrangements_scenario2) / Total_number_of_arrangements
Probability = (2 + 2(n-2)) / (2^n)
Therefore, the probability that there is at most one empty parking space between Mary and Tom's cars is (2 + 2(n-2)) / (2^n), which can be simplified to (2n) / (2^n), or simply n / (2^(n-1)).