This exercise is on probabilties and coincidence of shared birthdays. Complete parts (a) through (e) below. 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. (lgnore leap years and assume 365 days in a year) The first person can have any birthday, so they 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 birthday (Type whole numbers) b. If five people are selected at random, find the probability that they all have different birthdays. The probability that they all have different birthdays is $\square$ (Round to three decimal places as needed)

Step 1 ：Given that there are 365 days in a year, the probability that two people selected at random do not have the same birthday is calculated as follows: The first person can have a birthday on any of the 365 days, hence the probability is \(\frac{365}{365}\). For the second person to not have the same birthday, they must have one of the remaining 364 days, hence the probability is \(\frac{364}{365}\). Therefore, the total probability is \(\frac{365}{365} \cdot \frac{364}{365}\).

Step 2 ：To find the probability that five people selected at random all have different birthdays, we calculate the probability for each person. The first person can have a birthday on any of the 365 days, hence the probability is \(\frac{365}{365}\). For the second person to not have the same birthday, they must have one of the remaining 364 days, hence the probability is \(\frac{364}{365}\). This pattern continues for all five people, hence the total probability is \(\frac{365}{365} \cdot \frac{364}{365} \cdot \frac{363}{365} \cdot \frac{362}{365} \cdot \frac{361}{365}\).

Step 3 ：Calculating the above expression, we find that the probability that five people, selected at random, all have different birthdays is approximately 0.973.

Step 4 ：Final Answer: The probability that five people, selected at random, all have different birthdays is approximately \(\boxed{0.973}\).