Chapter 11

G.41
1. In how many possible ways can you land on jail (just visiting) on your first turn when playing a Monopoly game?
2. Is it possible to make it from GO to Park Place on your first roll of the dice in a Monopoly game? If so, what is the probability not only that would happen, but also that you would obtain a 2 on your next roll to complete a set (a monopoly).

G.42 Birthday problem:

1. Experiment: Consider the birthdates of some famous mathematicians:

   Abel    August 5, 1802

   Cardano    September 24, 1501

   Descartes    March 31, 1596

   Euler    April 15, 1707

   Fermat    August 18, 1602

   Galois    October 25, 1811

   Gauss    April 30, 1777

   Newton    December 25, 1642

   Pascal    June 19, 1623

   Riemann    September 17, 1826

   Add to this list the birthdates of the members of your class. But before you compile this list, guess the probability that at least two people in this group will have exactly the same birthday (not counting the year). Be sure to make your guess before finding out the birthdates of your classmates. The answer, of course, depends on the number of people on the list. Ten mathematicians are listed and you may have 20 people in your class, giving 30 names on the list.

2. Find the probability of at least one birthday match among 3 randomly selected people. (See Example 4, Section 0.4.)
3. Find the probability of at least one birthday match among 23 randomly selected people. Have each person in your group pick 23 names at random from a biographical dictionary or a Who's Who, and verify empirically the probability you calculated.
4. Draw a graph showing the probability of a birthday match given a group of n people. How many people are necessary for the probability actually to reach 1?
5. In the previous parts of this problem we interpreted two people having the same birthday as meaning at least 2 have the same birthday (see Example 4, Section .4). We now refine this idea. Find the following probabilities for a group of 5 randomly selected people:

   Exactly 2 of the 5 have the same birthday.
   Exactly 3 have the same birthday.
   Exactly 4 have the same birthday.
   All 5 have the same birthday.
   There are exactly two pairs sharing (a different) birthday.
   There is a full house of birthdays (that is, three share one birthday, and two share another).
   Show that the questions of this problem account for all the possibilities; that is, show that the sum of the probabilities for all of these possibilities is the same as for the original birthday problem involving 5 persons: What is the probability of a birthday match among 5 randomly selected people?

G43. Consider the following classroom activity. Suppose the floor consists of square tiles 9 in. on each side. The players will toss a circular disk onto the floor. If the disk comes to rest on the edge of any tile, the player loses $1. Otherwise, the player wins $1. What is the probability of winning if the disk is:

a. a dime b. a quarter c. a disk with a diameter of 4 in.

d. Now, the real question: What size should the disk be so that the probability that the player wins is 0.45?