318
Sets, Combinatorics, and Probability
Exercises 51−56 relate to the well-known “birthday problem.” Consider a room containing n persons, each of
whom has a birthday equally likely to fall on day 1−365 of the year (ignoring leap years).
51. What is the size of the sample space of all possible assignments of birthdays to people?
52. Let E be the event that no 2 persons in the room have the same birthday. Find an expression for P(E).
53. Let B be the event that 2 or more persons in the room share the same birthday. Find an expression for P(B).
54. Let C be the event that exactly 2 persons in the room share the same birthday. Find an expression for P(C ).
55. Which has the larger probability, B or C, and why?
56. Use a spreadsheet or calculator to determine that 23 is the minimum number of people in the room to give
a probability of least 1/2 that 2 or more people share the same birthday.
Exercises 57−62 refer to the gambling game of roulette. A roulette wheel contains 18 black slots and 18 red
slots that are numbered (not sequentially) 1−36. There are two green slots numbered 0 and 00. The dealer spins
the wheel and spins a ball into the wheel in the opposite direction. Bets are made on which slot the ball will fall
into as the wheel slows to a stop. The ball is equally likely to fall into any slot.
57. What is the size of the sample space?
58. What is the probability that the ball lands in a red slot (a “red” bet)?
59. What is the probability that the ball lands in a specific numbered slot (a “straight up” bet)?
60. What is the probability that the ball lands on one of three specific numbers (a “street” bet)?
61. What is the probability that the ball lands on one of four specific numbers (a “corner” bet)?
62. What is the probability that the ball lands on an odd number three times in a row?
63. A woman in Lake Havasu City, Arizona, gave birth to twin girls on September 22, 2006. This happened
to be the same date as the woman’s birthday—and the same date as her mother’s birthday. What is the
probability that three generations share the same birthday?
64. The NCAA men’s Division I college basketball championship (known as “March Madness”) is a singleelimination tournament held in March of each year. The tournament begins with 68 teams. The structure
of the tournament is as follows; the loser of each game is eliminated:
What is the probability of correctly picking the championship team? (Assume that all teams are equally
likely to win their respective games.)
name of the Round
number of teams
number of games
The First Four
8
4
Round of 64
64
32
Round of 32
32
16
Sweet Sixteen
16
8
Elite Eight
8
4
Final Four
4
2
Championship Game
2
1
Sets, Combinatorics, and Probability
Exercises 51−56 relate to the well-known “birthday problem.” Consider a room containing n persons, each of
whom has a birthday equally likely to fall on day 1−365 of the year (ignoring leap years).
51. What is the size of the sample space of all possible assignments of birthdays to people?
52. Let E be the event that no 2 persons in the room have the same birthday. Find an expression for P(E).
53. Let B be the event that 2 or more persons in the room share the same birthday. Find an expression for P(B).
54. Let C be the event that exactly 2 persons in the room share the same birthday. Find an expression for P(C ).
55. Which has the larger probability, B or C, and why?
56. Use a spreadsheet or calculator to determine that 23 is the minimum number of people in the room to give
a probability of least 1/2 that 2 or more people share the same birthday.
Exercises 57−62 refer to the gambling game of roulette. A roulette wheel contains 18 black slots and 18 red
slots that are numbered (not sequentially) 1−36. There are two green slots numbered 0 and 00. The dealer spins
the wheel and spins a ball into the wheel in the opposite direction. Bets are made on which slot the ball will fall
into as the wheel slows to a stop. The ball is equally likely to fall into any slot.
57. What is the size of the sample space?
58. What is the probability that the ball lands in a red slot (a “red” bet)?
59. What is the probability that the ball lands in a specific numbered slot (a “straight up” bet)?
60. What is the probability that the ball lands on one of three specific numbers (a “street” bet)?
61. What is the probability that the ball lands on one of four specific numbers (a “corner” bet)?
62. What is the probability that the ball lands on an odd number three times in a row?
63. A woman in Lake Havasu City, Arizona, gave birth to twin girls on September 22, 2006. This happened
to be the same date as the woman’s birthday—and the same date as her mother’s birthday. What is the
probability that three generations share the same birthday?
64. The NCAA men’s Division I college basketball championship (known as “March Madness”) is a singleelimination tournament held in March of each year. The tournament begins with 68 teams. The structure
of the tournament is as follows; the loser of each game is eliminated:
What is the probability of correctly picking the championship team? (Assume that all teams are equally
likely to win their respective games.)
name of the Round
number of teams
number of games
The First Four
8
4
Round of 64
64
32
Round of 32
32
16
Sweet Sixteen
16
8
Elite Eight
8
4
Final Four
4
2
Championship Game
2
1