Question

Suppose that the birthday of each of three people is equally likely to be any one of the 365 days of the year, independently of others. Let B(sub ij) denote the event that person i has the same birthday as person j, where the labels i and j may be 1, 2, or 3.

Are the events B(sub 12) and B(sub 23) independent?

Answer 1

The events B12 and B23 are independent because the occurrence of one does not affect the probability of the other occurring; each person's birthday is chosen independently of the others.

Step-by-step explanation:

The question asks whether the events B12 and B23 are independent. To determine if two events are independent, the probability of one event occurring should not affect the probability of the other event occurring. In the context of birthdays, B12 refers to the event where person 1 and person 2 have the same birthday, and B23 refers to the event where person 2 and person 3 have the same birthday.

For two events to be independent, the probability of both events occurring together should be the product of their individual probabilities. In the case of birthdays, assuming each date is equally likely, the probability that two people share a birthday is ⅛ (or 1/365). The probability of both B12 and B23 happening would then be ⅛² if they are independent.

However, if person 1 and person 2 share a birthday, it doesn't affect the probability that person 2 and person 3 also share a birthday (which also remains ⅛), because person 3's birthday is independent of person 1's. Therefore, B12 and B23 are independent events because the probability of B23 occurring is not affected by whether or not B12 has occurred.