Probability problems day 3

1. All 4500 Dartmouth students own a phone or a laptop. Say that 2000 own a laptop and 4000 own a phone. How many own a phone and a laptop?

2. Three people take off their coats for a party. Then, everyone picks up a coat at random, with each coat equally likely for each person.

   1. Let \(A_1\), \(A_2\), and \(A_3\) be the events that the three people get their coats back, respectively. Write an event which means ‘at least one person gets their coat back’.

   2. Compute the probability that at least one person gets their coat back using inclusion-exclusion.

   3. Compute the probability that no one gets their coat back.

3. Pick an integer from \(\{1, 2, 3, \dots, 100\}\) at random. What is the probability that the integer is not divisible by 2 or 5?

Hint. Let \(A\) be the event that the integer is divisible by \(2\), and \(B\) the event that the integer is divisible by 5. You want \(P((A \cup B)^C)\). Try complementing!