Final answer:
The probability that at least one guest will show up and a guest needing to be walked is 87.5%, calculated by subtracting the probability of no guests arriving (12.5%) from 1.
Step-by-step explanation:
The question at hand involves calculating the probability that at least one guest will be walked if the hotel has sold its last available room, and three guests have not checked in by 11:00 pm, with the individual probability of each guest not showing up being 50%. To find the overall probability of needing to walk a guest, we need to consider the different combinations of guests arriving or not arriving and then summing the probabilities of the scenarios where at least one guest arrives.
These possible scenarios are: no guests arrive, one guest arrives, two guests arrive, and all three guests arrive. Because the guests' arrivals are independent events, we can calculate the probabilities for these scenarios. The probability of needing to walk a guest is the probability that at least one guest arrives since all rooms are booked. This can be represented as 1 minus the probability that no guests arrive. Since each guest has a 50% chance of not showing up, the probability that no guest shows up is (0.5 x 0.5 x 0.5) or 0.125.
Therefore, the probability that at least one guest will show up (and a guest needing to be walked) is 1 - 0.125, which gives us 0.875 or 87.5%.