Page 472 - Discrete Mathematics and Its Applications

7.1 An Introduction to Discrete Probability 451

Exercises

1. What is the probability that a card selected at random 23. What is the probability that a positive integer not exceed-
from a standard deck of 52 cards is an ace? ing 100 selected at random is divisible by 5 or 7?
2. What is the probability that a fair die comes up six when 24. Find the probability of winning a lottery by selecting the
it is rolled? correct six integers, where the order in which these inte-
3. What is the probability that a randomly selected integer gers are selected does not matter, from the positive inte-
chosen from the first 100 positive integers is odd? gers not exceeding
4. What is the probability that a randomly selected day of a a) 30. b) 36. c) 42. d) 48.
leap year (with 366 possible days) is in April? 25. Find the probability of winning a lottery by selecting the
5. What is the probability that the sum of the numbers on correct six integers, where the order in which these inte-
two dice is even when they are rolled? gers are selected does not matter, from the positive inte-
gers not exceeding
6. What is the probability that a card selected at random
from a standard deck of 52 cards is an ace or a heart? a) 50. b) 52. c) 56. d) 60.
7. What is the probability that when a coin is flipped six 26. Find the probability of selecting none of the correct six in-
times in a row, it lands heads up every time? tegers in a lottery, where the order in which these integers
are selected does not matter, from the positive integers not
8. What is the probability that a five-card poker hand con-
tains the ace of hearts? exceeding
a) 40. b) 48. c) 56. d) 64.
9. What is the probability that a five-card poker hand does
not contain the queen of hearts? 27. Find the probability of selecting exactly one of the correct
six integers in a lottery, where the order in which these
10. What is the probability that a five-card poker hand con-
tains the two of diamonds and the three of spades? integers are selected does not matter, from the positive
integers not exceeding
11. What is the probability that a five-card poker hand con-
tains the two of diamonds, the three of spades, the six of a) 40. b) 48. c) 56. d) 64.
hearts, the ten of clubs, and the king of hearts? 28. In a superlottery, a player selects 7 numbers out of the
first 80 positive integers. What is the probability that a
12. What is the probability that a five-card poker hand con-
tains exactly one ace? person wins the grand prize by picking 7 numbers that
are among the 11 numbers selected at random by a com-
13. What is the probability that a five-card poker hand con- puter.
tains at least one ace?
29. In a superlottery, players win a fortune if they choose the
14. What is the probability that a five-card poker hand con-
eight numbers selected by a computer from the positive
tains cards of five different kinds?
integers not exceeding 100. What is the probability that a
15. What is the probability that a five-card poker hand con- player wins this superlottery?
tains two pairs (that is, two of each of two different kinds 30. What is the probability that a player of a lottery wins
and a fifth card of a third kind)? the prize offered for correctly choosing five (but not six)
16. What is the probability that a five-card poker hand con- numbers out of six integers chosen at random from the
tains a flush, that is, five cards of the same suit? integers between 1 and 40, inclusive?
17. What is the probability that a five-card poker hand con- 31. Suppose that 100 people enter a contest and that different
tains a straight, that is, five cards that have consecutive winners are selected at random for first, second, and third
kinds? (Note that an ace can be considered either the low- prizes. What is the probability that Michelle wins one of
est card of an A-2-3-4-5 straight or the highest card of a these prizes if she is one of the contestants?
10-J-Q-K-A straight.) 32. Suppose that 100 people enter a contest and that different
18. What is the probability that a five-card poker hand con- winners are selected at random for first, second, and third
tains a straight flush, that is, five cards of the same suit of prizes. What is the probability that Kumar, Janice, and
consecutive kinds? Pedro each win a prize if each has entered the contest?
∗ 19. What is the probability that a five-card poker hand con- 33. What is the probability that Abby, Barry, and Sylvia win
tains cards of five different kinds and does not contain a the first, second, and third prizes, respectively, in a draw-
flush or a straight? ing if 200 people enter a contest and
20. What is the probability that a five-card poker hand con- a) no one can win more than one prize.
tains a royal flush, that is, the 10, jack, queen, king, and b) winning more than one prize is allowed.
ace of one suit? 34. What is the probability that Bo, Colleen, Jeff, and Rohini
21. What is the probability that a fair die never comes up an win the first, second, third, and fourth prizes, respectively,
even number when it is rolled six times? in a drawing if 50 people enter a contest and
22. What is the probability that a positive integer not exceed- a) no one can win more than one prize.
ing 100 selected at random is divisible by 3? b) winning more than one prize is allowed.