Question

a. if two people are selected at random, the probability that they do not have the same birthday (day and month) is \\(\frac{365}{365} \cdot \frac{364}{365}\\). explain why this is so. (ignore leap years and assume 365 days in a year.)
the first person can have any birthday, so they can have a birthday on \\(square\\) of the 365 days. in order for the second person to not have the same birthday they must have one of the \\(square\\) remaining birthdays. (type whole numbers.)

Explanation:

Step1: Analyze the first - person's birthday

The first person can have a birthday on any of the 365 days. So, the number of ways the first person can choose a birthday is \(n_1 = 365\) (since there are 365 possible days).

Step2: Analyze the second - person's birthday

For the two people not to have the same birthday, when the first person has already chosen a birthday (in 365 ways), the second person must choose a birthday from the remaining \(365 - 1=364\) days. The number of ways the second person can choose a non - matching birthday is \(n_2 = 364\).

Step3: Calculate the total number of ways to choose birthdays for two people

The total number of ways two people can choose birthdays (without any restrictions) is \(N=365\times365\) (by the fundamental counting principle: if one event can occur in \(m\) ways and another independent event can occur in \(n\) ways, then the two events together can occur in \(m\times n\) ways). Here, \(m = 365\) (ways for the first person) and \(n = 365\) (ways for the second person).

Step4: Calculate the probability

The probability \(P\) that two people do not have the same birthday is given by the formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). The number of favorable outcomes (different birthdays) is \(365\times364\), and the total number of outcomes is \(365\times365\). So, \(P=\frac{365\times364}{365\times365}=\frac{364}{365}\)

Answer:

The first person can have a birthday on \(365\) of the \(365\) days. In order for the second person to not have the same birthday they must have one of the \(364\) remaining birthdays.