Problem 12
Suppose you hash \(n \log n\) items into \(n\) buckets. What is the expected maximum number of items in a bucket?
Problem 12
A die is made of a cube with a square painted on one side, a circle on two sides, and a triangle on three sides. If the die is rolled twice, what is the probability that the two shapes you see on top are the same?
Problem 12
In a family consisting of a mother, father, and two children of different ages, what is the probability that the family has two girls, given that one of the children is a girl? What is the probability that the children are both boys, given that the older child is a boy?
Problem 13
Are the following two events equally likely? Event 1 consists of drawing an ace and a king when you draw two cards from among the 13 spades in a deck of cards. Event 2 consists of drawing an ace and a king when you draw two cards from the whole deck.
Problem 13
You are a contestant on the TV game show Let’s Make a Deal. In this game show, there are three curtains. Behind one of the curtains is a new car, and behind the other two are cans of Spam. You get to pick one of the curtains. After you pick one of the curtains, the emcee, Monty Hall, who we assume knows where the car is, reveals what is behind one of the curtains that you did not pick, showing you some cans of Spam. He then asks you if you would like to switch your choice of curtain. Should you switch? Why or why not? Please answer this question carefully. You have all the tools needed to answer it, but several math Ph.D.s are on record (in Parade magazine) giving the wrong answer.
Problem 13
The fact that \(\lim _{n \rightarrow \infty}(1+1 / n)^{n}=e\) (where \(n\) varies over integers) is a consequence of the fact that \(\lim _{h \rightarrow 0}(1+h)^{1 / h}=e\) (where \(h\) varies over real numbers). Thus, if \(h\) varies over negative real numbers but approaches 0 , the limit still exists and equals \(e\). What does this tell you about \(\lim _{n \rightarrow-\infty}(1+1 / n)^{n} ?\) Using this and rewriting \((1-1 / n)^{n}\) as \((1+1 /-n)^{n}\), show that $$ \lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{n}=\frac{1}{e} $$
Problem 14
Given a random variable \(X\), how does the variance of \(c X\) relate to that of \(X\) ?
Problem 14
In as many ways as you can, prove that $$ \sum_{i=0}^{n} i\left(\begin{array}{l} n \\ i \end{array}\right)=n 2^{n-1} $$
Problem 14
There is a retired professor who used to love to go into a probability class of 30 or more students and announce, "I will give even money odds that there are two people in this classroom with the same birthday." With 30 students in the room, what is the probability that all have different birthdays? What is the minimum number of students that must be in the room so that the professor has probability at least \(1 / 2\) of winning the bet? What is the probability that he wins his bet if there are 50 students in the room? Does this probability make sense to you? (There is no wrong answer to this last question!) Explain why or why not. (A programmable calculator, spreadsheet, computer program, or computer algebra system will be helpful in this problem.)
Problem 15
Which is more likely, or are both equally likely? a. Drawing an ace and a king when you draw two cards from among the 13 spades, or drawing an ace and a king when you draw two cards from an ordinary deck of 52 playing cards? b. Drawing an ace and a king of the same suit when you draw two cards from a deck, or drawing an ace and a king when you draw two cards from among the 13 spades?
