Three friends were all born in the same year, which was not a leap year. Assuming that each friend’s birthday is independent of the others’ birthdays and that every day of the year is equally likely to be a birthday, find the probability that the friends all have the same birthday.
Let’s think about what we know. The three friends are all born in the same year, and that year was not a leap year. This tells us that the days we’re considering will be a total of 365 days. We also know that the birthdays are independent of one another. We also know that, for independent events, the probability of 𝐴 and 𝐵 occurring is equal to the probability of 𝐴 times the probability of 𝐵. But this is where we need to think carefully. We have to carefully label the events that are actually occurring here.
In this problem, the event is a shared birthday. We want to let event 𝐴 be the probability that friend one and friend two share a birthday. That would mean we would need a second event 𝐵, where friend one and friend three share a birthday. It would be really tempting to add a third event comparing friend two and friend three. However, if it is true that friend one and friend two share a birthday and it is true that friend one and friend three share a birthday, then all three friends will share a birthday. And this means we only have two independent events to consider.
The chances of friend one and friend two sharing a birthday is one out of 365. And the same thing is true for the chances of friend one and friend three sharing a birthday, one day out of 365, which means the probability that all three friends share a birthday is one out of 133,225.