Exercises 9.6 Exercises
1.
A ball is drawn randomly from a jar that contains 6 red balls, 2 white balls, and 5 yellow balls. Find the probability of the given event.
A red ball is drawn
A white ball is drawn
\(\displaystyle \frac{6}{13}\)
\(\displaystyle \frac{2}{13}\)
2.
Suppose you write each letter of the alphabet on a different slip of paper and put the slips into a hat. What is the probability of drawing one slip of paper from the hat at random and getting:
A consonant
A vowel
3.
A group of people were asked if they had run a red light in the last year. 150 responded "yes", and 185 responded "no". Find the probability that if a person is chosen at random, they have run a red light in the last year.
4.
In a survey, 205 people indicated they prefer cats, 160 indicated they prefer dots, and 40 indicated they don’t enjoy either pet. Find the probability that if a person is chosen at random, they prefer cats.
5.
Compute the probability of tossing a six-sided die (with sides numbered 1 through 6) and getting a 5.
6.
Compute the probability of tossing a six-sided die and getting a 7.
7.
Giving a test to a group of students, the grades and gender are summarized below.
| A | B | C | Total | |
|---|---|---|---|---|
| Male | 8 | 18 | 13 | 39 |
| Female | 10 | 4 | 12 | 26 |
| Total | 18 | 22 | 25 | 65 |
If one student was chosen at random, find the probability that the student was female.
What is the probability that a student chosen at random did NOT earn a C?
\(\displaystyle \frac{26}{65}=0.4\)
\(\displaystyle \frac{40}{65}=0.615\)
8.
The table below shows the number of credit cards owned by a group of individuals.
| Zero | One | Two or More | Total | |
|---|---|---|---|---|
| Male | 9 | 5 | 19 | 33 |
| Female | 18 | 10 | 20 | 48 |
| Total | 27 | 15 | 39 | 81 |
If one person was chosen at random, find the probability that the person had no credit cards.
What is the probability that a person chosen at random has at least one credit card?
9.
Compute the probability of tossing a six-sided die and getting an even number.
10.
Compute the probability of tossing a six-sided die and getting a number less than 3.
11.
If you pick one card at random from a standard deck of cards, what is the probability it will be a King?
12.
If you pick one card at random from a standard deck of cards, what is the probability it will be a Diamond?
13.
Compute the probability of rolling a 12-sided die and getting a number other than 8.
14.
If you pick one card at random from a standard deck of cards, what is the probability it is not the Ace of Spades?
15.
A six-sided die is rolled twice. What is the probability of showing a 6 on both rolls?
16.
A fair coin is flipped twice. What is the probability of showing heads on both flips?
17.
A die is rolled twice. What is the probability of showing a 5 on the first roll and an even number on the second roll?
18.
Suppose that 21% of people own dogs. If you pick two people at random, what is the probability that they both own a dog?
19.
Suppose a jar contains 17 red marbles and 32 blue marbles. If you reach in the jar and pull out 2 marbles at random, find the probability that both are red.
20.
Suppose you write each letter of the alphabet on a different slip of paper and put the slips into a hat. If you pull out two slips at random, find the probability that both are vowels.
21.
Bert and Ernie each have a well-shuffled standard deck of 52 cards. They each draw one card from their own deck. Compute the probability that:
Bert and Ernie both draw an Ace.
Bert draws an Ace but Ernie does not.
Neither Bert nor Ernie draws an Ace.
Bert and Ernie both draw a heart.
Bert gets a card that is not a Jack and Ernie draws a card that is not a heart.
\(\displaystyle \frac{1}{169}\)
\(\displaystyle \frac{12}{169}\)
\(\displaystyle \frac{144}{169}\)
\(\displaystyle \frac{1}{16}\)
\(\displaystyle \frac{117}{169}\)
22.
Bert has a well-shuffled standard deck of 52 cards, from which he draws one card; Ernie has a 12-sided die, which he rolls at the same time Bert draws a card. Compute the probability that:
Bert gets a Jack and Ernie rolls a five.
Bert gets a heart and Ernie rolls a number less than six.
Bert gets a face card (Jack, Queen or King) and Ernie rolls an even number.
Bert gets a red card and Ernie rolls a fifteen.
Bert gets a card that is not a Jack and Ernie rolls a number that is not twelve.
23.
Compute the probability of drawing a King from a deck of cards and then drawing a Queen.
24.
Compute the probability of drawing two spades from a deck of cards.
25.
A math class consists of 25 students, 14 female and 11 male. Two students are selected at random to participate in a probability experiment. Compute the probability that
a male is selected, then a female.
a female is selected, then a male.
two males are selected.
two females are selected.
no males are selected.
\(\displaystyle \frac{77}{300}\)
\(\displaystyle \frac{77}{300}\)
\(\displaystyle \frac{11}{60}\)
\(\displaystyle \frac{91}{300}\)
\(\displaystyle \frac{91}{300}\)
26.
A math class consists of 25 students, 14 female and 11 male. Three students are selected at random to participate in a probability experiment. Compute the probability that
a male is selected, then two females.
a female is selected, then two males.
two females are selected, then one male.
three males are selected.
three females are selected.
27.
Giving a test to a group of students, the grades and gender are summarized below.
| A | B | C | Total | |
|---|---|---|---|---|
| Male | 8 | 18 | 13 | 39 |
| Female | 10 | 4 | 12 | 26 |
| Total | 18 | 22 | 25 | 65 |
If one student was chosen at random, find the probability that the student was female and earned an A.
Find the probability that a student chosen at random is female or earned a B.
\(\displaystyle \frac{2}{13}\)
\(\displaystyle \frac{44}{65}\)
28.
The table below shows the number of credit cards owned by a group of individuals.
| Zero | One | Two or More | Total | |
|---|---|---|---|---|
| Male | 9 | 5 | 19 | 33 |
| Female | 18 | 10 | 20 | 48 |
| Total | 27 | 15 | 39 | 81 |
If one person was chosen at random, find the probability that the person was male and had two or more credit cards.
Find the probability that a person chosen at random is male or has no credit cards.
29.
A jar contains 6 red marbles numbered 1 to 6 and 8 blue marbles numbered 1 to 8. A marble is drawn at random from the jar. Find the probability the marble is red or odd-numbered.
30.
A jar contains 4 red marbles numbered 1 to 4 and 10 blue marbles numbered 1 to 10. A marble is drawn at random from the jar. Find the probability the marble is blue or even-numbered.
31.
Compute the probability of drawing the King of hearts or a Queen from a deck of cards.
32.
Compute the probability of drawing a King or a heart from a deck of cards.
33.
A jar contains 5 red marbles numbered 1 to 5 and 8 blue marbles numbered 1 to 8. A marble is drawn at random from the jar. Find the probability the marble is
Even-numbered given that the marble is red.
Red given that the marble is even-numbered.
\(\displaystyle \frac{2}{5}\)
\(\displaystyle \frac{1}{3}\)
34.
A jar contains 4 red marbles numbered 1 to 4 and 8 blue marbles numbered 1 to 8. A marble is drawn at random from the jar. Find the probability the marble is
Odd-numbered given that the marble is blue.
Blue given that the marble is odd-numbered.
35.
Compute the probability of flipping a coin and getting heads, given that the previous flip was tails.
36.
Find the probability of rolling a “1” on a fair die, given that the last 3 rolls were all ones.
37.
Suppose a math class contains 25 students, 14 females (three of whom speak French) and 11 males (two of whom speak French). Compute the probability that a randomly selected student speaks French, given that the student is female.
38.
Suppose a math class contains 25 students, 14 females (three of whom speak French) and 11 males (two of whom speak French). Compute the probability that a randomly selected student is male, given that the student speaks French.
39.
A certain virus infects one in every 400 people. A test used to detect the virus in a person is positive 90% of the time if the person has the virus and 10% of the time if the person does not have the virus.
Find the probability that a person has the virus given that they have tested positive.
Find the probability that a person does not have the virus given that they test negative.
0.022 = 2.2%
0.9997 = 99.97%
40.
A certain virus infects one in every 2000 people. A test used to detect the virus in a person is positive 96% of the time if the person has the virus and 4% of the time if the person does not have the virus.
Find the probability that a person has the virus given that they have tested positive
Find the probability that a person does not have the virus given that they test negative.
41.
A certain disease has an incidence rate of 0.3%. If the false negative rate is 6% and the false positive rate is 4%, compute the probability that a person who tests positive actually has the disease.
42.
A certain disease has an incidence rate of 0.1%. If the false negative rate is 8% and the false positive rate is 3%, compute the probability that a person who tests positive actually has the disease.
43.
A certain group of symptom-free women between the ages of 40 and 50 are randomly selected to participate in mammography screening. The incidence rate of breast cancer among such women is 0.8%. The false negative rate for the mammogram is 10%. The false positive rate is 7%. If a the mammogram results for a particular woman are positive (indicating that she has breast cancer), what is the probability that she actually has breast cancer?
44.
About 0.01% of men with no known risk behavior are infected with HIV. The false negative rate for the standard HIV test 0.01% and the false positive rate is also 0.01%. If a randomly selected man with no known risk behavior tests positive for HIV, what is the probability that he is actually infected with HIV?
45.
A license plate contains 3 randomly selected letters following by three randomly selected numbers. What is the probability the three letters are QQQ?
46.
You are taking a true/false test with 5 questions. If you randomly guess the answer on all five questions, what is the probability that you will get 100% on the test?
47.
You own 16 CDs. You want to randomly arrange 5 of them in a CD rack. What is the probability that the rack ends up in alphabetical order?
48.
A jury pool consists of 27 people, 14 men and 13 women. Compute the probability that a randomly selected jury of 12 people is all male.
49.
In a lottery game, a player picks six numbers from 1 to 48. If 5 of the 6 numbers match those drawn, they player wins second prize. What is the probability of winning this prize?
50.
In a lottery game, a player picks six numbers from 1 to 48. If 4 of the 6 numbers match those drawn, they player wins third prize. What is the probability of winning this prize?
51.
Compute the probability that a 5-card poker hand is dealt to you that contains all hearts.
52.
Compute the probability that a 5-card poker hand is dealt to you that contains four Aces.
53.
A bag contains 3 gold marbles, 6 silver marbles, and 28 black marbles. Someone offers to play this game: You randomly select on marble from the bag. If it is gold, you win $3. If it is silver, you win $2. If it is black, you lose $1. What is your expected value if you play this game?
54.
A friend devises a game that is played by rolling a single six-sided die once. If you roll a 6, he pays you $3; if you roll a 5, he pays you nothing; if you roll a number less than 5, you pay him $1. Compute the expected value for this game. Should you play this game?
55.
In a lottery game, a player picks six numbers from 1 to 23. If the player matches all six numbers, they win 30,000 dollars. Otherwise, they lose $1. Find the expected value of this game.
56.
A game is played by picking two cards from a standard deck. If they are the same suit, then you win $5, otherwise you lose $1. What is the expected value of this game?
57.
A company estimates that 0.7% of their products will fail after the original warranty period but within 2 years of the purchase, with a replacement cost of $350. If they offer a 2 year extended warranty for $48, what is the company's expected value of each warranty sold?
58.
An insurance company estimates the probability of an earthquake in the next year to be 0.0013. The average damage done by an earthquake it estimates to be $60,000. If the company offers earthquake insurance for $100, what is their expected value of the policy?
59.
A magazine subscription is having a contest in which the price is $80,000. If the company receives 1 million entries, what is the expected value of an entry in the contest?
60.
In a game show, four prizes are hidden on a board that contains 20 spaces. One prize is worth $15,000, two are worth $5,000 and the other prize is worth $1,000. The remaining spaces contain no prizes. The game show host offers a guaranteed prize of $1,000 not to play this game. Use expected value to determine whether the contestant should choose the guaranteed $1,000 or play the game?
61.
A game involves drawing a single card from a standard deck of 52 cards. If an ace is drawn you ge $0.50; if a face card is drawn you get $0.25; if the 2 of clubs is drawn you get $1. If the cost of playing is $0.10, should you play the game?
62.
The SAT is a multiple-choice test. Each question has five possible choices. The test taker must select one answer for each question or leave the question blank. One point is award for each correct answer and -1/4 point is subtracted for each incorrect answer. (No points are added or subtracted for questions that are left blank.)
What is the expected value of randomly guessing the answer to a question
If you can eliminate one of the choices, what is the expected value of randomly guessing among the remaining four choices? Is it a good strategy to guess? Why or why not?
63.
A construction company is planning to bid on a building contract. The bid cost the company $1200. the probability that the bid is accepted is 0.2. If the bid is accepted the company will make $40,000 minus the cost of the bid. Find the expected value of the making the bid.