Final answer:
The probability that all three people will get off on the 4th floor is 1/512. The probability they all get off on the same floor is 1/64, that at least two get off on the same floor is 235/256, and that exactly two get off on the same floor is 21/512.
Step-by-step explanation:
Part (a): The probability that all three people get off on the 4th floor is the product of the probabilities of each individual person getting off on the 4th floor. Since there are 8 floors to choose from (2-9), and the choice is random, the probability for each person is 1/8. The probability for all three is then (1/8) x (1/8) x (1/8) which equals 1/512.
Part (b): To find the probability that all three get off on the same floor, we need to consider that the first person can choose any floor (8 possibilities), and the next two must choose the same floor as the first. The probability is hence 8 x (1/8) x (1/8), simplifying to 1/64.
Part (c): For at least two to get off on the same floor, we can find the probability of the complement event where all three get off on different floors and subtract it from 1. The probability of all three selecting different floors is (8/8) x (7/8) x (6/8), simplifying to 42/512. Therefore, the probability of at least two on the same floor is 1 - 42/512, which is 470/512, or 235/256.
Part (d): To find the probability that exactly two people get off on the same floor, we can use a combination of selecting 2 out of 3 people and multiply by the probability of those two choosing the same floor and the third one choosing a different floor. There are 3C2 ways to choose 2 people from 3, which is 3, and the probability is 3 x (1/8) x (1/8) x (7/8), equating to 21/512.