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Inside Collection:

Collection by: Rupinder Sekhon. E-mail the author

# More Probability: Homework

Module by: UniqU, LLC. E-mail the author

Summary: This chapter covers additional principles of probability. After completing this chapter students should be able to: find the probability of a binomial experiment; find the probabilities using Bayes' Formula; find the expected value or payoff in a game of chance; find the probabilities using tree diagrams.

## BINOMIAL PROBABILITY

Do the following problems using the binomial probability formula.

### Exercise 1

A coin is tossed ten times. Find the probability of getting six heads and four tails.

0.2051

### Exercise 2

A family has three children. Find the probability of having one boy and two girls.

### Exercise 3

What is the probability of getting three aces(ones) if a die is rolled five times?

0.0322

### Exercise 4

A baseball player has a .250 batting average. What is the probability that he will have three hits in five times at bat?

### Exercise 5

A basketball player has an 80% chance of sinking a basket on a free throw. What is the probability that he will sink at least three baskets in five free throws?

0.9421

### Exercise 6

With a new flu vaccination, 85% of the people in the high risk group can go through the entire winter without contracting the flu. In a group of six people who were vaccinated with this drug, what is the probability that at least four will not get the flu?

### Exercise 7

A transistor manufacturer has known that 5% of the transistors produced are defective. What is the probability that a batch of twenty five will have two defective?

0.2305

### Exercise 8

It has been determined that only 80% of the people wear seat belts. If a police officer stops a car with four people, what is the probability that at least one person will not be wearing a seat belt?

### Exercise 9

What is the probability that a family of five children will have at least three boys?

0.5

### Exercise 10

What is the probability that a toss of four coins will yield at most two heads?

### Exercise 11

A telemarketing executive has determined that for a particular product, 20% of the people contacted will purchase the product. If 10 people are contacted, what is the probability that at most 2 will buy the product?

0.6778

### Exercise 12

To the problem: "Five cards are dealt from a deck of cards, find the probability that three of them are kings," the following incorrect answer was offered by a student.

5C31/13312/1325C31/13312/132 size 12{5C3 left (1/"13" right ) rSup { size 8{3} } left ("12"/"13" right ) rSup { size 8{2} } } {}
(1)

What change would you make in the wording of the problem for the given answer to be correct?

## BAYES' FORMULA

Use both tree diagrams and Bayes' formula to solve the following problems.

### Exercise 13

Jar I contains five red and three white marbles, and Jar II contains four red and two white marbles. A jar is picked at random and a marble is drawn. Draw a tree diagram below, and find the following probabilities.

1. Pmarble is redPmarble is red size 12{P left ("marble is red" right )} {}

2. PIt came from Jar II given that the marble drawn is whitePIt came from Jar II given that the marble drawn is white size 12{P left ("It came from Jar II given that the marble drawn is white" right )} {}

3. PRed Jar IPRed Jar I size 12{P left ("Red " \lline " Jar I" right )} {}

1. 0.6458
2. 0.4706
3. 0.625

### Exercise 14

In Mr. Symons' class, if a person does his homework most days, his chance of passing the course is 90%. On the other hand, if a person does not do his homework most days his chance of passing the course is only 20%. Mr. Symons claims that 80% of his students do their homework on a regular basis. If a student is chosen at random from Mr. Symons' class, find the following probabilities.

1. Pthe student passes the coursePthe student passes the course size 12{P left ("the student passes the course" right )} {}

2. Pthe student did homework the student passes the coursePthe student did homework the student passes the course size 12{P left ("the student did homework " \lline " the student passes the course" right )} {}

3. Pthe student passes the course the student did homeworkPthe student passes the course the student did homework size 12{P left ("the student passes the course " \lline " the student did homework" right )} {}

### Exercise 15

A city has 60% Democrats, and 40% Republicans. In the last mayoral election, 60% of the Democrats voted for their Democratic candidate while 95% of the Republicans voted for their candidate. Which party's mayor runs city hall?

#### Solution

The Republican Party

### Exercise 16

In a certain population of 48% men and 52% women, 56% of the men and 8% of the women are color-blind.

1. What percent of the people are color-blind?

2. If a person is found to be color-blind, what is the probability that the person is a male?

### Exercise 17

A test for a certain disease gives a positive result 95% of the time if the person actually carries the disease. However, the test also gives a positive result 3% of the time when the individual is not carrying the disease. It is known that 10% of the population carries the disease. If the test is positive for a person, what is the probability that he or she has the disease?

0.7787

### Exercise 18

A person has two coins: a fair coin and a two-headed coin. A coin is selected at random, and tossed. If the coin shows a head, what is the probability that the coin is fair?

### Exercise 19

A computer company buys its chips from three different manufacturers. Manufacturer I provides 60% of the chips and is known to produce 5% defective; Manufacturer II supplies 30% of the chips and makes 4% defective; while the rest are supplied by Manufacturer III with 3% defective chips. If a chip is chosen at random, find the following probabilities.

1. Pthe chip is defectivePthe chip is defective size 12{P left ("the chip is defective" right )} {}

2. Pit came from Manufacturer II the chip is defectivePit came from Manufacturer II the chip is defective size 12{P left ("it came from Manufacturer II " \lline " the chip is defective" right )} {}

3. Pthe chip is defective it came from manufacturer IIIPthe chip is defective it came from manufacturer III size 12{P left ("the chip is defective " \lline " it came from manufacturer III" right )} {}

1. 0.045
2. 0.2667
3. 0.03

### Exercise 20

Lincoln Union High School District is made up of three high schools: Monterey, Fremont, and Kennedy, with an enrollment of 500, 300, and 200, respectively. On a given day, the percentage of students absent at Monterey High School is 6%, at Fremont 4%, and at Kennedy 5%. If a student is chosen at random, find the following probabilities. Hint: Convert the enrollments into percentages.

1. Pthe student is absentPthe student is absent size 12{P left ("the student is absent" right )} {}

2. Pthe student came from Kennedy the student is absentPthe student came from Kennedy the student is absent size 12{P left ("the student came from Kennedy " \lline " the student is absent" right )} {}

3. Pthe student is absent the student came from FremontPthe student is absent the student came from Fremont size 12{P left ("the student is absent " \lline " the student came from Fremont" right )} {}

## EXPECTED VALUE

Do the following problems using the expected value concepts learned in this section,

### Exercise 21

You are about to make an investment which gives you a 30% chance of making $60,000 and 70% chance of losing$ 30,000. Should you invest? Explain.

50 cents

### Exercise 24

A game involves rolling a single die. One receives the face value of the die in dollars. How much should one be willing to pay to roll the die to make the game fair?

### Exercise 25

In a European country, 20% of the families have three children, 40% have two children, 30% have one child, and 10% have no children. On average, how many children are there to a family?

1.7

### Exercise 26

A game involves drawing a single card from a standard deck. One receives 60 cents for an ace, 30 cents for a king, and 5 cents for a red card that is neither an ace nor a king. If the cost of each draw is 10 cents, should one play? Explain.

### Exercise 27

Hillview Church plans to raise money by raffling a television worth $500. A total of 3000 tickets are sold at$1 each. Find the expected value of the winnings for a person who buys a ticket in the raffle.

-83 cents

### Exercise 28

During her four years at college, Niki received A's in 30% of her courses, B's in 60% of her courses, and C's in the remaining 10%. If A=4A=4 size 12{A=4} {}, B=3B=3 size 12{B=3} {}, and C=2C=2 size 12{C=2} {}, find her grade point average.

### Exercise 29

Attendance at a Stanford football game depends upon which team Stanford is playing against. If the game is against U. C. Berkeley, the attendance will be 70,000; if it is against another California team, it will be 40,000; and if it is against an out of state team, it will be 30,000. If the probability of playing against U. C. Berkeley is 10%, against a California team 50% , and against an out of state team 40%, how many fans are expected to attend a game?

39,000

-96 cents

### Exercise 61

A roulette wheel consists of numbers 1 through 36, 0, and 00. If the wheel comes up an odd number you win a dollar, otherwise you lose a dollar. If you play the game ten times, what is your expectation?

### Exercise 67

Jar I contains 1 red and 3 white, and Jar II contains 2 red and 3 white marbles. A marble is drawn from Jar I and put in Jar II. Now if one marble is drawn from Jar II, what is the probability that it is a red marble?

#### Solution

3/83/8 size 12{3/8} {}
(5)

### Exercise 68

Let us suppose there are three traffic lights between your house and the school. The chance of finding the first light green is 60%, the second 50%, and the third 30%. What is the probability that on your way to school, you will find at least two lights green?

0.45

### Exercise 69

Sonya has just earned her law degree and is planning to take the bar exam. If her chance of passing the bar exam is 65% on each try, what is the probability that she will pass the exam in at least three tries?

0.957125

### Exercise 70

Every time Ken Griffey is at bat, his probability of getting a hit is .3, his probability of walking is .1, and his probability of being struck out is .4. If he is at bat three times, what is the probability that he will get two hits and one walk?

0.027

### Exercise 71

Jar I contains 4 marbles of which none are red, and Jar II contains 6 marbles of which 4 are red. Juan first chooses a jar and then from it he chooses a marble. After the chosen marble is replaced, Mary repeats the same experiment. What is the probability that at least one of them chooses a red marble?

#### Solution

5/95/9 size 12{5/9} {}
(6)

### Exercise 72

Andre and Pete are two tennis players with equal ability. Andre makes the following offer to Pete: We will not play more than four games, and anytime I win more games than you, I am declared a winner and we stop. Draw a tree diagram and determine Andre's probability of winning.

#### Solution

5/85/8 size 12{5/8} {}
(7)

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