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Chapter 7

Discrete Probability

Educators


Problem 1

What is the probability that a card selected at random from a standard deck of 52 cards is an ace?

Dalia R.
Numerade Educator

Problem 2

What is the probability that a fair die comes up six when it is rolled?

James C.
Numerade Educator

Problem 3

What is the probability that a randomly selected integer chosen from the first 100 positive integers is odd?

Dalia R.
Numerade Educator

Problem 4

What is the probability that a randomly selected integer chosen from the first 100 positive integers is odd?

James C.
Numerade Educator

Problem 5

What is the probability that the sum of the numbers on two dice is even when they are rolled?

Dalia R.
Numerade Educator

Problem 6

What is the probability that the sum of the numbers on two dice is even when they are rolled?

James C.
Numerade Educator

Problem 7

What is the probability that when a coin is flipped six times in a row, it lands heads up every time?

Dalia R.
Numerade Educator

Problem 8

What is the probability that a five-card poker hand contains the ace of hearts?

James C.
Numerade Educator

Problem 9

What is the probability that a five-card poker hand does not contain the queen of hearts?

Dalia R.
Numerade Educator

Problem 10

What is the probability that a five-card poker hand contains the two of diamonds and the three of spades?

James C.
Numerade Educator

Problem 11

What is the probability that a five-card poker hand contains the two of diamonds, the three of spades, the six of hearts, the ten of clubs, and the king of hearts?

Dalia R.
Numerade Educator

Problem 12

What is the probability that a five-card poker hand contains exactly one ace?

James C.
Numerade Educator

Problem 13

What is the probability that a five-card poker hand contains at least one ace?

Dalia R.
Numerade Educator

Problem 14

What is the probability that a five-card poker hand contains cards of five different kinds?

James C.
Numerade Educator

Problem 15

What is the probability that a five-card poker hand contains two pairs (that is, two of each of two different kinds and a fifth card of a third kind)?

Dalia R.
Numerade Educator

Problem 16

What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

James C.
Numerade Educator

Problem 17

What is the probability that a five-card poker hand contains a straight, that is, five cards that have consecutive kinds? (Note that an ace can be considered either the lowest card of an A $-2-3-4-5$ straight or the highest card of a $10-\mathrm{J}-\mathrm{Q}-\mathrm{K}-\mathrm{A}$ straight.)

Dalia R.
Numerade Educator

Problem 18

What is the probability that a five-card poker hand contains a straight flush, that is, five cards of the same suit of consecutive kinds?

James C.
Numerade Educator

Problem 19

What is the probability that a five-card poker hand contains cards of five different kinds and does not contain a flush or a straight?

Dalia R.
Numerade Educator

Problem 20

What is the probability that a five-card poker hand contains a royal flush, that is, the $10,$ jack, queen, king, and ace of one suit?

James C.
Numerade Educator

Problem 21

What is the probability that a fair die never comes up an even number when it is rolled six times?

Dalia R.
Numerade Educator

Problem 22

What is the probability that a positive integer not exceeding 100 selected at random is divisible by 3$?$

James C.
Numerade Educator

Problem 23

What is the probability that a positive integer not exceeding 100 selected at random is divisible by 5 or 7$?$

Dalia R.
Numerade Educator

Problem 24

Find the probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive inte- gers not exceeding
$$\begin{array}{llll}{\text { a) } 30 .} & {\text { b) } 36 .} & {\text { c) } 42 .} & {\text { d) } 48}\end{array}$$

James C.
Numerade Educator

Problem 25

Find the probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding
$$\begin{array}{llll}{\text { a) } 50 .} & {\text { b) } 52 .} & {\text { c) } 56 .} & {\text { d) } 60}\end{array}$$

Dalia R.
Numerade Educator

Problem 26

Find the probability of selecting none of the correct six integers in a lottery, where the order in which these integers are selected does not matter, from the positive integers not exceeding
$$\begin{array}{llll}{\text { a) } 40 .} & {\text { b) } 48} & {\text { c) } 56} & {\text { d) } 64}\end{array}$$

James C.
Numerade Educator

Problem 27

Find the probability of selecting exactly one of the correct six integers in a lottery, where the order in which these integers are selected does not matter, from the positive integers not exceeding
$$\begin{array}{llll}{\text { a) } 40 .} & {\text { b) } 48} & {\text { c) } 56} & {\text { d) } 64}\end{array}$$

Dalia R.
Numerade Educator

Problem 28

In a superlottery, a player selects 7 numbers out of the first 80 positive integers. What is the probability that a person wins the grand prize by picking 7 numbers that are among the 11 numbers selected at random by a computer.

James C.
Numerade Educator

Problem 29

In a superlottery, players win a fortune if they choose the eight numbers selected by a computer from the positive integers not exceeding $100 .$ What is the probability that a player wins this superlottery?

Dalia R.
Numerade Educator

Problem 30

What is the probability that a player of a lottery wins the prize offered for correctly choosing five (but not six) numbers out of six integers chosen at random from the integers between 1 and $40,$ inclusive?

James C.
Numerade Educator

Problem 31

Suppose that 100 people enter a contest and that different winners are selected at random for first, second, and third prizes. What is the probability that Michelle wins one of these prizes if she is one of the contestants?

Dalia R.
Numerade Educator

Problem 32

Suppose that 100 people enter a contest and that different winners are selected at random for first, second, and third prizes. What is the probability that Kumar, Janice, and Pedro each win a prize if each has entered the contest?

James C.
Numerade Educator

Problem 33

What is the probability that Abby, Barry, and Sylvia win the first, second, and third prizes, respectively, in a drawing if 200 people enter a contest and
a) no one can win more than one prize.
b) winning more than one prize is allowed.

Dalia R.
Numerade Educator

Problem 34

What is the probability that Bo, Colleen, Jeff, and Rohini win the first, second, third, and fourth prizes,
respectively, in a drawing if 50 people enter a contest and
a) no one can win more than one prize.
b) winning more than one prize is allowed.

James C.
Numerade Educator

Problem 35

In roulette, a wheel with 38 numbers is spun. Of these, 18 are red, and 18 are black. The other two numbers, which are neither black nor red, are 0 and $00 .$ The probability that when the wheel is spun it lands on any particular number is 1$/ 38$ .
a) What is the probability that the wheel lands on a red number?
b) What is the probability that the wheel lands on a black number twice in a row?
c) What is the probability that the wheel lands on 0 or 00$?$ do?
d) What is the probability that in five spins the wheel never lands on either 0 or 00$?$
e) What is the probability that the wheel lands on one of the first six integers on one spin, but does not land on any of them on the next spin?

Dalia R.
Numerade Educator

Problem 36

Which is more likely: rolling a total of 8 when two dice are rolled or rolling a total of 8 when three dice are rolled?

James C.
Numerade Educator

Problem 37

Which is more likely: rolling a total of 9 when two dice are rolled or rolling a total of 9 when three dice are rolled?

Dalia R.
Numerade Educator

Problem 38

A player in the Mega Millions lottery picks five different integers between 1 and $70,$ inclusive, and a sixth integer between 1 and $25,$ inclusive, which may duplicate one of the earlier five integers. The player wins the jackpot if all six numbers match the numbers drawn.
a) What is the probability that a player wins the jackpot?
b) What is the probability that a player wins $\$ 1,000,000$ , the prize for matching the first five numbers, but not the sixth number, drawn?
c) What is the probability that a player wins $\$ 500$ , the prize for matching exactly four of the first five numbers, but not the sixth number, drawn?
d) What is the probability that a player wins $\$ 10,$ the prize for matching exactly three of the first five numbers but not the sixth number drawn, or for matching exactly two of the first five numbers and the sixth number drawn?

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Problem 39

When a player buys a Mega Millions ticket in many states see Exercise 38 , the player can also buy the Megaplier, which multiplies the size of a prize other than a jackpot by a multiplier ranging from two to five. The Megaplier is drawn using a pool of 15 balls, with five marked 2 $\mathrm{X}$ , six marked 3 $\mathrm{X}$ , three marked 4 $\mathrm{X}$ , and one marked 5 $\mathrm{X}$ , where each ball has the same likelihood of being drawn. Find the probability that a player who buys a Mega Mil- lions ticket and the Megaplier wins
a) $\$ 5,000,000 ?$ (The only way to do this is to match the first five numbers drawn but not the sixth number drawn, with Megaplier $5 \mathrm{X} .$ )
b) $\$ 30,000 ?$ (The only way to do this is to match exactly four of the first five numbers drawn and the sixth number drawn, with Megaplier 3X.)
c) $\$ 20 ?$ (The three ways to do this are to match exactly three of the first five numbers drawn, but not the sixth number drawn, or exactly two of the first five numbers and the sixth number, with Megaplier $2 \mathrm{X},$ or to match exactly one of the first five numbers and the sixth number, with Megaplier 5 $\mathrm{X}$ .)
d) $\$ 8 ?$ (The two ways to do this are to match exactly one of the first five numbers and the sixth number drawn, with a multiplier of $2 \mathrm{X},$ or to match the sixth number but none of the first five numbers, with Megaplier $4 \mathrm{X} . )$

Dalia R.
Numerade Educator

Problem 40

A player in the Powerball lottery picks five different integers between 1 and 69 , inclusive, and a sixth integer between 1 and $26,$ which may duplicate one of the earlier five integers. The player wins the jackpot if all six numbers match the numbers drawn.
a) What is the probability that a player wins the jackpot?
b) What is the probability that a player wins $\$ 1,000,000$ , which is the prize for matching the first five numbers, but not the sixth number, drawn?
c) What is the probability that a player wins $\$ 100$ by matching exactly three of the first five and the sixth numbers drawn, or four of the first five numbers, but not the sixth number, drawn?
d) What is the probability that a player wins a prize of $\$ 4,$ which is the prize when the player matches the sixth number, and either one or none of the first five numbers drawn?

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Problem 41

A player in the Powerball lottery (see Exercise 40$)$ can purchase the Power Play option. When this option has been purchased, prizes other than the jackpot are multiplied by a multiplier, chosen using a random number generator with weighted values for the different multipliers. When the jackpot is more than $\$ 150,000,000$ , the weighted values are 24 for $2 \mathrm{X}, 13$ for $3 \mathrm{X}, 3$ for $4 \mathrm{X},$ and 2 for 5 $\mathrm{X}$ . When the jackpot does not exceed $\$ 150,000,000,$ the weighted values are 24 for $2 \mathrm{X}, 13$ for $3 \mathrm{X}, 3$ for $4 \mathrm{X}, 2$ for 5 $\mathrm{X}$ , and 1 for 10 $\mathrm{X}$ . All non-jackpot prizes are multiplied by the multiplier chosen, except for the $\$ 1,000,000$ prize, which is doubled when the Power Play option is in effect regardless of the multiplier chosen. What is the probability that a play who has purchased a Powerball ticket and Power Play wins
a) $\$ 2,000,000,$ if the jackpot is more than $\$ 150,000,000 ?$
b) $\$ 2,000,000,$ if the jackpot does not exceed $\$ 150,000,000 ?$
c) $\$ 1000$ , if the jackpot does not exceed $\$ 150,000,000 ?$ (The two ways to do this are for the Power Play multiplier to be 10 $\mathrm{X}$ , and to match either exactly four of the first five numbers but not the sixth number drawn, or exactly three of the first five numbers and the sixth number drawn.)
d) $\$ 12,$ if the jackpot is more than $\$ 150,000,000 ?$ (The two ways to do this are for the Power Play multiplier to be 3 $\mathrm{X}$ and to match the sixth number and either one or none of the first five numbers drawn.)

Dalia R.
Numerade Educator

Problem 42

Two events $E_{1}$ and $E_{2}$ are called independent if $p\left(E_{1} \cap E_{2}\right)=p\left(E_{1}\right) p\left(E_{2}\right) .$ For each of the following pairs of events, which are subsets of the set of all possible outcomes when a coin is tossed three times, determine whether or not they are independent.
a) $E_{1} :$ tails comes up with the coin is tossed the first time; $E_{2} :$ heads comes up when the coin is tossed the second time.
b) $E_{1} :$ the first coin comes up tails; $E_{2} :$ two, and not three, heads come up in a row.
c) $E_{1} :$ the second coin comes up tails; $E_{2} :$ two, and not three, heads come up in a row. (We will study independence of events in more depth in Section $7.2 . )$

James C.
Numerade Educator

Problem 43

Explain what is wrong with the statement that in the Monty Hall Three-Door Puzzle the probability that the
prize is behind the first door you select and the probability that the prize is behind the other of the two doors that Monty does not open are both $1 / 2,$ because there are two doors left.

Dalia R.
Numerade Educator

Problem 44

Suppose that instead of three doors, there are four doors in the Monty Hall puzzle. What is the probability that you win by not changing once the host, who knows what is behind each door, opens a losing door and gives you the chance to change doors? What is the probability that you win by changing the door you select to one of the two remaining doors among the three that you did not select?

James C.
Numerade Educator

Problem 45

This problem was posed by the Chevalier de Méré and was solved by Blaise Pascal and Pierre de Fermat.
a) Find the probability of rolling at least one six when a fair die is rolled four times.
b) Find the probability that a double six comes up at least once when a pair of dice is rolled 24 times. Answer the query the Chevalier de Méré made to Pascal asking whether this probability was greater than 1$/ 2$ .
c) Is it more likely that a six comes up at least once when a fair die is rolled four times or that a double six comes up at least once when a pair of dice is rolled 24 times?

Dalia R.
Numerade Educator