Section 4.6 Probability
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38. In the Daily4 game, four numbers between 0 and 9 will be drawn in succession (repetitions allowed). The
player marks four numbers on a game card and has a choice of how to play, straight (the player’s numbers
will match the four numbers drawn in exact order) or box (the player’s numbers will match the four drawn
in any order).
a. What is the size of the sample space?
b. What is the probability of a straight?
c. What is the probability of a box if four distinct numbers are drawn?
d. What is the probability of a box if two of the numbers drawn are the same?
e. What is the probability of a box if two distinct pairs of numbers are drawn?
f. What is the probability of a box if three of the numbers drawn are the same?
39. In the Cash5 game, five different numbers between 1 and 39 will be drawn in succession. The player
marks five different numbers between 1 and 39 on a game card.
a. What is the size of the sample space?
b. What is the probability of matching all five numbers in any order?
c. What is the probability of matching exactly four of the five numbers in any order?
d. What is the probability of matching exactly three of the five numbers in any order?
40. In the Powerball game, five different numbers between 1 and 59 will be drawn in succession, and then one
number (the Powerball number) between 1 and 35 will be drawn. The player marks five different numbers
between 1 and 59 and one number between 1 and 35 on a game card.
a. What is the size of the sample space?
b. What is the probability of matching all five numbers in any order plus matching the Powerball number?
c. What is the probability of matching none of the five numbers but matching the Powerball number?
Exercises 41−50 concern 5-card poker hands from a standard 52-card deck. A standard deck has 13 cards from
each of 4 suits (clubs, diamonds, hearts, spades). The 13 cards have face value 2 through 10, jack, queen, king,
or ace. Each face value is a “kind” of card. The jack, queen, and king are “face cards.” (See Exercises 33−41
of Section 4.4. for the definitions of terms.)
41. What is the probability of a royal straight flush?
42. What is the probability of a straight flush that is not a royal straight flush?
43. What is the probability of four of a kind?
44. What is the probability of a full house?
45. What is the probability of a flush that is not a straight flush or a royal straight flush?
46. What is the probability of a straight that is not a straight flush or a royal straight flush?
47. What is the probability of three of a kind?
48. What is the probability of two pairs?
49. What is the probability of a pair?
50. Explain why, in poker,
a. a straight flush beats a full house.
b. four of a kind beats a straight.
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38. In the Daily4 game, four numbers between 0 and 9 will be drawn in succession (repetitions allowed). The
player marks four numbers on a game card and has a choice of how to play, straight (the player’s numbers
will match the four numbers drawn in exact order) or box (the player’s numbers will match the four drawn
in any order).
a. What is the size of the sample space?
b. What is the probability of a straight?
c. What is the probability of a box if four distinct numbers are drawn?
d. What is the probability of a box if two of the numbers drawn are the same?
e. What is the probability of a box if two distinct pairs of numbers are drawn?
f. What is the probability of a box if three of the numbers drawn are the same?
39. In the Cash5 game, five different numbers between 1 and 39 will be drawn in succession. The player
marks five different numbers between 1 and 39 on a game card.
a. What is the size of the sample space?
b. What is the probability of matching all five numbers in any order?
c. What is the probability of matching exactly four of the five numbers in any order?
d. What is the probability of matching exactly three of the five numbers in any order?
40. In the Powerball game, five different numbers between 1 and 59 will be drawn in succession, and then one
number (the Powerball number) between 1 and 35 will be drawn. The player marks five different numbers
between 1 and 59 and one number between 1 and 35 on a game card.
a. What is the size of the sample space?
b. What is the probability of matching all five numbers in any order plus matching the Powerball number?
c. What is the probability of matching none of the five numbers but matching the Powerball number?
Exercises 41−50 concern 5-card poker hands from a standard 52-card deck. A standard deck has 13 cards from
each of 4 suits (clubs, diamonds, hearts, spades). The 13 cards have face value 2 through 10, jack, queen, king,
or ace. Each face value is a “kind” of card. The jack, queen, and king are “face cards.” (See Exercises 33−41
of Section 4.4. for the definitions of terms.)
41. What is the probability of a royal straight flush?
42. What is the probability of a straight flush that is not a royal straight flush?
43. What is the probability of four of a kind?
44. What is the probability of a full house?
45. What is the probability of a flush that is not a straight flush or a royal straight flush?
46. What is the probability of a straight that is not a straight flush or a royal straight flush?
47. What is the probability of three of a kind?
48. What is the probability of two pairs?
49. What is the probability of a pair?
50. Explain why, in poker,
a. a straight flush beats a full house.
b. four of a kind beats a straight.