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# Chapter 14. Probability: Homework

### SAMPLE SPACES AND PROBABILITY

In problems 1 - 6, write a sample space for the given experiment.

A die is rolled.

A penny and a nickel are tossed.

A die is rolled, and a coin is tossed.

*,*2H

*,*3H

*,*4H

*,*5H

*,*6H

*,*1T

*,*2T

*,*3T

*,*4T

*,*5T

*,*6T}

Three coins are tossed.

Two dice are rolled.

1 | 2 | 3 | 4 | 5 | 6 | |

1 | (1, 1) | (1, 2) | (1, 3) | (1, 4) | (1, 5) | (1, 6) |

2 | (2, 1) | (2, 2) | (2, 3) | (2, 4) | (2, 5) | (2, 6) |

3 | (3, 1) | (3, 2) | (3, 3) | (3, 4) | (3, 5) | (3, 6) |

4 | (4, 1) | (4, 2) | (4, 3) | (4, 4) | (4, 5) | (4, 6) |

5 | (5, 1) | (5, 2) | (5, 3) | (5, 4) | (5, 5) | (5, 6) |

6 | (6, 1) | (6, 2) | (6, 3) | (6, 4) | (6, 5) | (6, 6) |

A jar contains four marbles numbered 1, 2, 3, and 4. Two marbles are drawn.

In problems 7 - 12, a card is selected from a deck. Find the following probabilities.

*P*(an ace)

*P*(a red card)

*P*(a club)

*P*(a face card)

*P*(a jack or spade)

*P*(a jack and a spade)

A jar contains 6 red, 7 white, and 7 blue marbles. If a marble is chosen at random, find the following probabilities.

*P*(red)

*P*(white)

*P*(red or blue)

*P*(red and blue)

Consider a family of three children. Find the following probabilities.

*P*(two boys and a girl)

*P*(at least one boy)

*P*(children of both sexes)

*P*(at most one girl)

Two dice are rolled. Find the following probabilities.

*P*(the sum of the dice is 5)

*P*(the sum of the dice is 8)

*P*(the sum is 3 or 6)

*P*(the sum is more than 10)

A jar contains four marbles numbered 1, 2, 3, and 4. If two marbles are drawn, find the following probabilities.

*P*(the sum of the number is 5)

*P*(the sum of the numbers is odd)

*P*(the sum of the numbers is 9)

*P*(one of the numbers is 3)

### MUTUALLY EXCLUSIVE EVENTS AND THE ADDITION RULE

Determine whether the following pair of events are mutually exclusive.

*A*={A person earns more than $25,000}

*B*={A person earns less than $20,000}

A card is drawn from a deck.

*C*={It is a King}

*D*={It is a heart}.

A die is rolled.

*E*={An even number shows}

*F*={A number greater than 3 shows}

Two dice are rolled.

*G*={The sum of dice is 8}

*H*={One die shows a 6}

Three coins are tossed.

*I*={Two heads come up}

*J*={At least one tail comes up}

A family has three children.

*K*={First born is a boy}

*L*={The family has children of both sexes}

Use the addition rule to find the following probabilities.

A card is drawn from a deck, and the events
*C* and
*D* are as follows:

*C*={It is a king}

*D*={It is a heart}

Find
*P*(*C* or *D*).

A die is rolled, and the events
*E* and
*F* are as follows:

*E*={An even number shows}

*F*={A number greater than 3 shows}

Find
*P*(*E* or *F*).

Two dice are rolled, and the events
*G* and
*H* are as follows:

*G*={The sum of dice is 8}

*H*={Exactly one die shows a 6}

Find
*P*(*G* or *H*).

Three coins are tossed, and the events
*I* and
*J* are as follows:

*I*={Two heads come up}

*J*={At least one tail comes up}

Find
*P*(*I* or *J*).

At De Anza college, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?

This quarter, there is a 50% chance that Jason will pass Accounting, a 60% chance that he will pass English, and 80% chance that he will pass at least one of these two courses. What is the probability that he will pass both Accounting and English?

The following table shows the distribution of Democratic and Republican U.S. Senators by gender.

MALES(M) | FEMALES(F) | TOTAL | |

DEMOCRATS(D) | 39 | 4 | 43 |

REPUBLICANS(R) | 52 | 5 | 57 |

TOTALS | 91 | 9 | 100 |

Use this table to determine the following probabilities.

*P*(

*M*and

*D*)

*P*(

*F*and

*R*)

*P*(

*M*or

*D*)

*P*(

*F*or

*R*

^{C})

*P*(

*M*

^{C}or

*R*)

*P*(

*M*or

*F*)

Again, use the addition rule to determine the following probabilities.

If
*P*(*E*)=.5 and
*P*(*F*)=.4 and
*E* and
*F* are mutually exclusive, find
*P*(*E* and *F*).

If
*P*(*E*)=.4 and
*P*(*F*)=.2 and
*E* and
*F* are mutually exclusive, find
*P*(*E* or *F*).

If
*P*(*E*)=.3 and
*P*(*E* or *F*)=.6 and
*P*(*E* and *F*)=.2, find
*P*(*F*).

If
*P*(*E*)=.4,
*P*(*F*)=.5 and
*P*(*E* or *F*)=.7, find
*P*(*E* and *F*).

### CALCULATING PROBABILITIES USING TREE DIAGRAMS AND COMBINATIONS

Two apples are chosen from a basket containing five red and three yellow apples. Draw a tree diagram below, and find the following probabilities.

*P*(both red)

*P*(one red, one yellow)

*P*(both yellow)

*P*(First red and second yellow)

A basket contains six red and four blue marbles. Three marbles are drawn at random. Find the following probabilities using the method shown in Example 13.2. Do not use combinations.

*P*(All three red)

*P*(two red, one blue)

*P*(one red, two blue)

*P*(first red, second blue, third red)

Three marbles are drawn from a jar containing five red, four white, and three blue marbles. Find the following probabilities using combinations.

*P*(all three red)

*P*(two white and 1 blue)

*P*(none white)

*P*(at least one red)

A committee of four is selected from a total of 4 freshmen, 5 sophomores, and 6 juniors. Find the probabilities for the following events.

At least three freshmen.

No sophomores.

All four of the same class.

Not all four from the same class.

Exactly three of the same class.

More juniors than freshmen and sophomores combined.

Five cards are drawn from a deck. Find the probabilities for the following events.

Two hearts, two spades, and one club.

*,*960

A flush of any suit(all cards of a single suit).

A full house of nines and tens(3 nines and 2 tens).

*,*960

Any full house.

A pair of nines and tens.

*,*960

Two pairs

Do the following birthday problems.

If there are five people in a room, what is the probability that no two have the same birthday?

If there are five people in a room, what is the probability that at least two people have the same birthday?

### CONDITIONAL PROBABILITY

Do the following problems using the conditional probability formula: .

A card is drawn from a deck. Find the conditional probability of
*P*(a queen ∣ a face card).

A card is drawn from a deck. Find the conditional probability of
*P*(a queen ∣ a club).

A die is rolled. Find the conditional probability that it shows a three if it is known that an odd number has shown.

If
*P*(*A*)=.3 and
*P*(*B*)=.4, and
*P*(*A* and *B*)=.12, find the following.

*P*(*A*∣*B*)*P*(*B*∣*A*)

The following table shows the distribution of Democratic and Republican U.S. Senators by gender.

MALE(M) | FEMALE(F) | TOTAL | |

DEMOCRATS(D) | 39 | 4 | 43 |

REPUBLICANS(R) | 52 | 5 | 57 |

TOTALS | 91 | 9 | 100 |

Use this table to determine the following probabilities:

*P*(

*M*∣

*D*)

*P*(

*D*∣

*M*)

*P*(

*F*∣

*R*)

*P*(

*R*∣

*F*)

Do the following conditional probability problems.

At De Anza College, 20% of the students take Finite Math, 30% take History, and 5% take both Finite Math and History. If a student is chosen at random, find the following conditional probabilities.

He is taking Finite Math given that he is taking History.

He is taking History assuming that he is taking Finite Math.

1/6

1/4

At a college, 60% of the students pass Accounting, 70% pass English, and 30% pass both of these courses. If a student is selected at random, find the following conditional probabilities.

He passes Accounting given that he passed English.

He passes English assuming that he passed Accounting.

If
*P*(*F*)=.4 and
*P*(*E*∣*F*)=.3, find
*P*(*E* and *F*).

If
*P*(*E*)=.3, and
*P*(*F*)=.3, and
*E* and
*F* are mutually exclusive, find
*P*(*E*∣*F*).

If
*P*(*E*)=.6 and
*P*(*E* and *F*)=.24, find
*P*(*F*∣*E*).

If
*P*(*E* and *F*)=.04 and
*P*(*E*∣*F*)=.1, find
*P*(*F*).

Consider a family of three children. Find the following probabilities.

*P*(two boys∣first born is a boy)

*P*(all girls ∣ at least one girl is born)

*P*(children of both sexes ∣ first born is a boy)

*P*(all boys ∣ there are children of both sexes)

### INDEPENDENT EVENTS

The distribution of the number of fiction and non-fiction books checked out at a city's main library and at a smaller branch on a given day is as follows.

MAIN(M) | BRANCH(B) | TOTAL | |

FICTION(F) | 300 | 100 | 400 |

NON-FICTION(N) | 150 | 50 | 200 |

TOTALS | 450 | 150 | 600 |

Use this table to determine the following probabilities:

*P*(

*F*)

*P*(

*M*∣

*F*)

*P*(

*N*∣

*B*)

Is the fact that a person checks out a fiction book independent of the main library?

For a two-child family, let the events
*E*,
*F*, and
*G* be as follows.

*E*: The family has at least one boy
*F*: The family has children of both sexes
*G*: The family's first born is a boy

Find the following.

*P*(*E*)*P*(*F*)*P*(*E*∩*F*)Are

*E*and*F*independent?

3/4

2/4

2/4

no

Find the following.

*P*(*F*)*P*(*G*)*P*(*F*∩*G*)Are

*F*and*G*independent?

Do the following problems involving independence.

If
*P*(*E*)=.6,
*P*(*F*)=.2, and
*E* and
*F* are independent, find
*P*(*E* and *F*).

If
*P*(*E*)=.6,
*P*(*F*)=.2, and
*E* and
*F* are independent, find
*P*(*E* or *F*).

If
*P*(*E*)=.9,
*P*(*F*∣*E*)=.36, and
*E* and
*F* are independent, find
*P*(*F*).

If
*P*(*E*)=.6,
*P*(*E* or *F*)=.08, and
*E* and
*F* are independent, find
*P*(*F*).

In a survey of 100 people, 40 were casual drinkers, and 60 did not drink. Of the ones who drank, 6 had minor headaches. Of the non-drinkers, 9 had minor headaches. Are the events "drinkers" and "had headaches" independent?

It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?

John's probability of passing statistics is 40%, and Linda's probability of passing the same course is 70%. If the two events are independent, find the following probabilities.

*P*(both of them will pass statistics)*P*(at least one of them will pass statistics)

28/100

82/100

Jane is flying home for the Christmas holidays. She has to change planes twice on the way home. There is an 80% chance that she will make the first connection, and a 90% chance that she will make the second connection. If the two events are independent, find the following probabilities.

*P*(Jane will make both connections)*P*(Jane will make at least one connection)

For a three-child family, let the events
*E*,
*F*, and
*G* be as follows.

*E*: The family has at least one boy
*F*: The family has children of both sexes
*G*: The family's first born is a boy

Find the following.

*P*(*E*)*P*(*F*)*P*(*E*∩*F*)Are

*E*and*F*independent?

7/8

6/8

6/8

no

Find the following.

*P*(*F*)*P*(*G*)*P*(*F*∩*G*)Are

*F*and*G*independent?

### CHAPTER REVIEW

Two dice are rolled. Find the probability that the sum of the dice is

four

five

3/36

4/36

A jar contains 3 red, 4 white, and 5 blue marbles. If a marble is chosen at random, find the following probabilities:

*P*(red or blue)*P*(not blue)

8/12

7/12

A card is drawn from a standard deck. Find the following probabilities:

*P*(a jack or a king)*P*(a jack or a spade)

8/52

16/52

A basket contains 3 red and 2 yellow apples. Two apples are chosen at random. Find the following probabilities:

*P*(one red, one yellow)*P*(at least one red)

3/5

9/10

A basket contains 4 red, 3 white, and 3 blue marbles. Three marbles are chosen at random. Find the following probabilities:

*P*(two red, one white)*P*(first red, second white, third blue)*P*(at least one red)*P*(none red)

3/20

1/20

5/6

1/6

Given a family of four children. Find the following probabilities:

*P*(All boys)*P*(1 boy and 3 girls)

1/16

1/4

Consider a family of three children. Find the following:

*P*(children of both sexes ∣ first born is a boy)*P*(all girls ∣ children of both sexes)

3/4

0

Mrs. Rossetti is flying from San Francisco to New York. On her way to the San Francisco Airport she encounters heavy traffic and determines that there is a 20% chance that she will be late to the airport and will miss her flight. Even if she makes her flight, there is a 10% chance that she will miss her connecting flight at Chicago. What is the probability that she will make it to New York as scheduled?

At a college, twenty percent of the students take history, thirty percent take math, and ten percent take both. What percent of the students take at least one of these two courses?

In a T-maze, a mouse may run to the right (R) or may run to the left (L). A mouse goes up the maze three times, and events
*E* and
*F* are described as follows:

*E*: Runs to the right on the first trial

*F*: Runs to the left two consecutive times

Determine whether the events
*E* and
*F* are independent.

A college has found that 20% of its students take advanced math courses, 40% take advanced English courses and 15% take both advanced math and advanced English courses. If a student is selected at random, what is the probability that

he is taking English given that he is taking math?

he is taking math or English?

3/4

0.45

If there are 35 students in a class, what is the probability that at least two have the same birthday?

A student feels that her probability of passing accounting is .62, of passing mathematics is .45, and her passing accounting or mathematics is .85. Find the probability that the student passes both accounting and math.

There are nine judges on the U. S. Supreme Court of which five are conservative and four liberal. This year the court will act on six major cases. What is the probability that out of six cases the court will favor the conservatives in at least four?

Five cards are drawn from a deck. Find the probability of obtaining

four cards of a single suit

two cards of one suit, two of another suit, and one from the remaining

a pair(e.g. two aces and three other cards)

a straight flush(five in a row of a single suit but not a royal flush)

111540/2598960

949104/2598960

1349088/2598960

36/2598960

The following table shows a distribution of drink preferences by gender.

Coke(C) | Pepsi(P) | Seven Up(S) | TOTALS | |

Males(M) | 60 | 50 | 22 | 132 |

Females(F) | 50 | 40 | 18 | 108 |

TOTALS | 110 | 90 | 40 | 240 |

The events
*M*,
*F*,
*C*,
*P* and
*S* are defined as Male, Female, Coca Cola, Pepsi, and Seven Up, respectively. Find the following:

*P*(*F*∣*S*)*P*(*P*∣*F*)*P*(*C*∣*M*)Are the events

*F*and*S*mutually exclusive?Are the events

*F*and*S*independent?

9/20

10/27

15/33

11/20

no

yes

At a clothing outlet 20% of the clothes are irregular, 10% have at least a button missing and 4% are both irregular and have a button missing. If Martha found a dress that has a button missing, what is the probability that it is irregular?

A trade delegation consists of four Americans, three Japanese and two Germans. Three people are chosen at random. Find the following probabilities:

*P*(two Americans and one Japanese)*P*(at least one American)*P*(One of each nationality)*P*(no German)

3/14

37/42

2/7

35/84

A coin is tossed three times, and the events
*E* and
*F* are as follows.

*E*: It shows a head on the first toss
*F*: Never turns up a tail

Are the events
*E* and
*F* independent?

If
*P*(*E*)=.6 and
*P*(*F*)=.4 and
*E* and
*F* are mutually exclusive, find
*P*(*E* and *F*).

If
*P*(*E*)=.5 and
*P*(*F*)=.3 and
*E* and
*F* are independent, find
.

If
*P*(*F*)=.9 and
*P*(*E*∣*F*)=.36 and
*E* and
*F* are independent, find
*P*(*E*).

If
*P*(*E*)=.4 and
*P*(*E* or *F*)=.9 and
*E* and
*F* are independent, find
*P*(*F*).

If
*P*(*E*)=.4 and
*P*(*F*∣*E*)=.5, find
*P*(*E* and *F*).

If
*P*(*E*)=.6 and
*P*(*E* and *F*)=.3, find
*P*(*F*∣*E*).

If
*P*(*E*)=.3 and
*P*(*F*)=.4 and
*E* and
*F* are independent, find
*P*(*E*∣*F*).