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## Educators

DG

### Problem 1

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

Dalia R.

### Problem 2

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

James C.

### Problem 3

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

Dalia R.

### Problem 4

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

James C.

### Problem 5

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

Dalia R.

### Problem 6

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

James C.

### 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.

### Problem 8

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

James C.

### Problem 9

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

Dalia R.

### Problem 10

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

James C.

### 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.

### Problem 12

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

James C.

### Problem 13

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

Dalia R.

### Problem 14

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

James C.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### Problem 21

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

Dalia R.

### Problem 22

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

James C.

### Problem 23

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

Dalia R.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

Dalia R.

### 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.

### 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.

James C.

### 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.

### 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.

### 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.

### 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? Check back soon! ### 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?

DG
Dustin G.