A group of 100 people are comparing their birthdays (as usual, assume their birthdays are independent and not on February 29, etc.). Find the expected number of pairs of people with the same birthday, and the expected number of days in the year on which at least two of these people were born.

To address the problem, let's break it down into two parts as specified:

Part 1: Expected Number of Pairs of People with the Same Birthday

We are given a group of 100 people, and we want to determine the expected number of pairs of people who share the same birthday. The key assumption here is that birthdays are independent and randomly distributed across the 365 days of the year.

Step 1: Probability of a Pair Sharing a Birthday

For each pair of people, the probability that they share the same birthday is 1365\frac{1}{365}. This is because there are 365 possible birthdays, and the second person has a 1365\frac{1}{365} chance of matching the birthday of the first person.

Step 2: Total Number of Pairs

The total number of ways to select a pair of people from the 100 people is given by the combination formula (1002)\binom{100}{2}. This is calculated as:

(1002)=100×992=4950\binom{100}{2} = \frac{100 \times 99}{2} = 4950

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