1. In a group of at least 24 randomly chosen people, what is the probability that some pair of them having the same birthday?| 2. Mr. Smith works on the 13th floor of a 15 floor building. The only elevator moves continuously through floors 1, 2, ..., 15, 14, . .., 2, 1, 2, ..., except that it stops on a floor on which the button has been pressed. Assume that time spent loading and unloading passengers is very small compared to the travelling time. Mr. Smith complains that at 5pm, when he wants to go home, the elevator almost always goes up when it stops on his floor. What is the explanation? Now assume that the building has n elevators, which move independently. Compute the proportion of time the first elevator on Mr. Smith's floor moves up. 3. Say you're on a game show where there are three doors. Behind two of the doors, there are goats. Behind one of the doors, there is a brand new car. The host says that once you pick a door, he'll open one of the doors you didn't pick to reveal a goat. Then, you have the option of either staying with your door or switching to the last unopened door. Will you switch the door and why