(Solved): CASE: Probability That at Least Two People in the Same Room Have the Same Birthday Suppose that ...

CASE: Probability That at Least Two People in the Same Room Have the Same Birthday Suppose that there are two people in a room. The probability that they share the same birthday (date, not necessarily year) is $1/365$, and the probability that they have different birthdays is $364/365$. To illustrate, suppose that you're in a room with one other person and that your birthday is July 1 . The probability that the other person does not have the same birthday is $364/365$ because there are 364 days in the year that are not July 1 . If a third person now enters the room, the probability that that person has a different birthday from the first two people in the room is $363/365$. Thus, the (joint) probability that three people in a room having different birthdays is ( $364/365$ ) ( $363/365$ ). You can continue this process for any number of people. Find the number of people in a room so that there is about (closest to) a $80\%$ probability that at least two have the same birthday. Hint 1: Calculate the probability that they don't have the same birthday (round your answers to four decimal places).. Hint 2: Excel users can employ the product function to calculate joint probabilities. Solutions: The probability that people have different birthdays is The probability that at least two people have the same birthday (the complement event) is