Deck 4: Probability

Question 1

Sometimes probabilities derived by the relative frequency method differ from the probabilities expected from classical probability methods. How does the law of large numbers apply in this situation?

Answer 1

The law of large numbers states that as an experiment is repeated again and again, the relative frequency probabilities tend to approach the actual probabilities expected from the classical approach.

Question 2

Compare the relative frequency formula for finding probabilities to the classical formula for finding probabilities. How are the two formulas similar and how are they different?
What special requirements does the classical approach have?

Answer 2

In the relative frequency formula, the probabilities are determined by conducting an experiment, counting the number of occurrences of the event, and creating the ratio of number of occurrences to number of times the experiment was conducted. In the classical approach, a sample space of all of the possible simple events is created and the number of successes is compared to the number of different simple events. In the classical approach, each outcome must be equally likely.

Question 3

Use a graphing calculator to develop a simulation of the given problem (100 trials). Describe the simulation, then estimate the probability based on its results.

-Assume that ten percent of us are left-handed. The probability of getting at least two left-handed people in a group of 7.

Answer 3

Answers may vary. Begin by representing left-handed people by the digit 0 and right-handed people by the other digits. Then use a calculator or computer program to generate 7 random integers. Count as a success if the last digit of two or more of the generated numbers is 0. Repeat for 100 trials. The estimated probability is the number of successes divided by 100. Estimated probability is about 0.26.

Question 4

Use a graphing calculator to develop a simulation of the given problem (100 trials). Describe the simulation, then estimate the probability based on its results.

-The probability of getting at least two boys in a family of seven children.

Question 5

Discuss why the relative frequency method of computing probabilities might be chosen over the classical method.

Question 6

Probabilities are useful in the decision-making process. Suppose a random sample of 152 students was surveyed regarding an instructor's teaching. Suppose 105 students rated the instructor either excellent or above average on lecture presentations, 96 students rated the instructor as giving difficult or very difficult assignments and tests. Would you take this instructor for a class? Discuss the influence of probabilities on making this decision.

Question 7

Use a graphing calculator to develop a simulation of the given problem (100 trials). Describe the simulation, then estimate the probability based on its results.

-The probability of getting a total of 15 when four dice are thrown.

Question 8

When asked about the probability that he would become a sumo wrestler, Sam replied "slim to none." Relate that phrase to a numeric probability and interpret his meaning.

Question 9

Compare probabilities and odds. How can you convert odds to probabilities?

Question 10

If a game were "fair," the payoff on a bet would be the same as the odds for the event. In one game, the odds for winning are 1:13. If the game were "fair," what would the payoff be for a $5 bet? Of course, games in casinos are designed to make a profit for the casino investors. Supposing the casino makes the payoff at 1:11 odds, what profit does the casino make on your winning bet?

Question 11

On an exam on probability concepts, Sue had an answer of $\frac { 13 } { 8 }$ for one problem. Explain how she knew that this result was incorrect.

Question 12

Give an example of events which are independent but not disjoint.

Question 13

Tomorrow Freddy is playing in an important soccer game. He reasons that there are three possible outcomes (they could win, tie, or lose) so the probability of winning must be 1/3. Is this reasoning correct? Why or why not?

Question 14

List two reasons it is better to sample without replacement when testing batches of products. When sampling without replacement, should you use the multiplication rule for independent or dependent events? Explain your answer.

Question 15

Describe an event whose probability of occurring is 1 and explain what that probability means. Describe an event whose probability of occurring is 0 and explain what that probability means.

Question 16

Use a graphing calculator to develop a simulation of the given problem (100 trials). Describe the simulation, then estimate the probability based on its results.

-The probability of getting exactly three girls in a family of five children.

Question 17

Which of the following cannot be a probability?

A) 0

B) $\frac { 1 } { 2 }$

C) $- 1$

D) 1

Question 18

Which of the following cannot be a probability?

A) $\frac { 2 } { 3 }$

B) $\frac { 3 } { 5 }$

C) $\frac { 5 } { 3 }$

D) $\frac { 1 } { 2 }$

Question 19

What important question must you answer before computing an "or" probability? How does the answer influence your computation?

Question 20

What important question must you answer before computing an "and" probability? How does the answer influence your computation?

Question 21

Find the indicated probability. Express your answer as a simplified fraction unless otherwise noted.

-The table below shows the soft drinks preferences of people in three age groups.
$$\begin{array} { r | c c c } & \text { cola } & \text { root beer } & \text { lemon-lime } \\\hline \text { under 21 years of age } & 40 & 25 & 20 \\\text { between 21 and 40 } & 35 & 20 & 30 \\\text { over 40 years of age } & 20 & 30 & 35\end{array}$$
If one of the 255 subjects is randomly selected, find the probability that the person is over 40 years of age.

A) $\frac { 1 } { 2 }$

B) $\frac { 2 } { 5 }$

C) $\frac { 3 } { 5 }$

D) $\frac { 1 } { 3 }$

Question 22

Provide a written description of the complement of the given event.
When several textbooks are edited, none of them are found to be free of errors.

A) All of the textbooks are free of errors.
B) One of the textbooks is free of errors.
C) At least one of the textbooks is free of errors.
D) At most one of the textbooks is free of errors.

Question 23

Describe the process for making a tree diagram and give an example.

Question 24

Discuss the range of possible values for probabilities. Give examples to support each.

Question 25

Under what circumstances can you sample without replacement and still use the multiplication rule for independence? Discuss population and sample size as you answer this question.

Question 26

Find the indicated probability.

-If you pick a card at random from a well shuffled deck, what is the probability that you get a face card or a spade?

A) $\frac { 1 } { 22 }$

B) $\frac { 9 } { 26 }$

C) $\frac { 25 } { 52 }$

D) $\frac { 11 } { 26 }$

Question 27

Use Bayes' theorem to find the indicated probability.

-3.6% of a population are infected with a certain disease. There is a test for the disease, however the test is not completely accurate. 95.3% of those who have the disease test positive. However 4.2% of those who do not have the disease also test positive (false positives). A person is randomly selected and tested for the disease. What is the probability that the person has the disease given that the test result is positive?

A) 0.847
B) 0.036
C) 0.541
D) 0.459

Question 28

Consider the following formulas: $${ } _ { n } \mathrm { P } _ { \mathrm { r } } = \frac { \mathrm { n } ! } { ( \mathrm { n } - \mathrm { r } ) ! } \text { and } \mathrm { n } _ { \mathrm { r } } = \frac { \mathrm { n } ! } { ( \mathrm { n } - \mathrm { r } ) ! \mathrm { r } ! }$$ Given the same values for n and r in each formula, which is the smaller value, P or C? How does this relate to the concept of counting the number of outcomes based on whether or not order is a criterion?

Question 29

Choose the one alternative that best completes the statement or answers the question.
Evaluate the expression.

- ${ } _ { 5 } \mathrm { P } _ { 4 }$

A) 15
B) 120
C) 0
D) 1

Question 30

Discuss the differences, both in applications and in the formulas, for combinations and permutations. Give an example of each.

Question 31

Discuss the advantages and disadvantages of odds.

Question 32

Determine whether the events are disjoint.

-Read a book by Mark Twain.
Read about Tom Sawyer.

A) Yes
B) No

Question 33

If $B$ denotes some event and $P ( B ) = \frac { 399 } { 400 }$ , what is the value of $\overline { P ( B ) }$ ? Is $B$ unusual? Is $\bar { B }$ unusual?

Question 34

Suppose a student is taking a 5-response multiple choice exam; that is, the choices are A, B, C, D, and E, with only one of the responses correct. Describe the complement method for determining the probability of getting at least one of the questions correct on the 15-question exam. Why would the complement method be the method of choice for this problem?

Question 35

How many 3-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7 if repetition of digits is not allowed?

A) 210
B) 5
C) 343
D) 6

Question 36

A tourist in France wants to visit 7 different cities. If the route is randomly selected, what is the probability that she will visit the cities in alphabetical order?

A) $\frac { 1 } { 7 }$

B) $\frac { 1 } { 49 }$

C) $\frac { 1 } { 5040 }$

D) 5040

Question 37

In a certain lottery, five different numbers between 1 and 21 inclusive are drawn. These are the winning numbers. To win the lottery, a person must select the correct 5 numbers in the same order In which they were drawn. What is the probability of winning?

A) $\frac { 1 } { 21 ! }$

B) $\frac { 1 } { 2,441,880 }$

C) $\frac { 120 } { 2,441,880 }$

D) $\frac { 1 } { 120 }$

Question 38

Use a graphing calculator to develop a simulation of the given problem (100 trials). Describe the simulation, then estimate the probability based on its results.

-The probability that when 75 people are surveyed, at least five of them have the same birth date. (Ignore leap years).

Question 39

Define disjoint events and independent events. Give an example of each.

Question 40

Interpret the symbol $\mathrm { P } ( \mathrm { B } \mid \mathrm { A } )$ and explain what is meant by the expression. What do we know if $\mathrm { P } ( \mathrm { B } \mid \mathrm { A } )$ is not the same as $\mathrm { P } ( \mathrm { B } )$ ?

Question 41

Provide a written description of the complement of the given event.
Of ten adults, at least one of them has high blood pressure.

A) Nine of the adults have high blood pressure.
B) At most one of the adults has high blood pressure.
C) All of the adults have high blood pressure.
D) None of the adults have high blood pressure.

Question 42

Answer the question, considering an event to be "unusual" if its probability is less than or equal to 0.05.

-Assume that a study of 300 randomly selected school bus routes showed that 281 arrived on time. Is it "unusual" for a school bus to arrive late?

A) Yes
B) No

Question 43

What is the probability of an event that is certain to occur?

A) 1
B) 0.99
C) 0.5
D) 0.95

Question 44

What is the probability of an impossible event?

A) 0.1
B) 0
C) - 1
D) 1

Question 45

How many ways can an IRS auditor select 6 of 13 tax returns for an audit?

A) 4,826,809
B) 720
C) 1,235,520
D) 1716

Question 46

Find the indicated probability. Express your answer as a simplified fraction unless otherwise noted.

-The table below shows the soft drinks preferences of people in three age groups.
$$\begin{array} { r | c c c } & \text { cola } & \text { root beer } & \text { lemon-lime } \\\hline \text { under 21 years of age } & 40 & 25 & 20 \\\text { between 21 and 40 } & 35 & 20 & 30 \\\text { over 40 years of age } & 20 & 30 & 35\end{array}$$
If one of the 255 subjects is randomly selected, find the probability that the person is over 40 years of age given that they drink root beer.

A) $\frac { 5 } { 17 }$

B) $\frac { 6 } { 17 }$

C) $\frac { 2 } { 5 }$

D) None of the above is correct.

Question 47

Of the 54 people who answered "yes" to a question, 12 were male. Of the 46 people that answered "no" to the question, 12 were male. If one person is selected at random from the group, what is the probability that the person answered "yes" or was male?

A) 0.78
B) 0.24
C) 0.222
D) 0.66

Question 48

A class has 11 students who are to be assigned seating by lot. What is the probability that the students will be arranged in order from shortest to tallest? (Assume that no two students are the same height.)

A) 0.00000025
B) 0.00000028
C) 0.00000003
D) 0.1000

Question 49

Find the indicated probability.

-100 employees of a company are asked how they get to work and whether they work full time or part time. The figure below shows the results. If one of the 100 employees is randomly selected, find the probability of getting someone who carpools or someone who works full time.


1. Public transportation: 7 full time, 8 part time
2) Bicycle: 4 full time, 5 part time
3) Drive alone: 35 full time, 25 part time
4) Carpool: 8 full time, 8 part time

A) 0.26
B) 0.62
C) 0.58
D) 0.7

Question 50

Swinging Sammy Skor's batting prowess was simulated to get an estimate of the probability that Sammy will get a hit. Let $1 =$ HIT and $2 =$ OUT . The output from the simulation was as follows.
$\begin{array}{llcc}1&2&2&2&1&2&2&1&1&1&2&2&2&2&1&1&1&1&2&2&2&2&1&1&2&2&1&1&1&1&2&2&2&2&2&1&1&1&1\end{array}$
Estimate the probability that he makes an out.

A) 0.524
B) 0.782
C) 0.643
D) 0.621

Question 51

Express the indicated degree of likelihood as a probability value.
"Your mother could not have died two years before you were born."

A) 0.5
B) 1
C) 0.25
D) 0

Question 52

You are dealt two cards successively (without replacement) from a shuffled deck of 52 playing cards. Find the probability that both cards are black. Express your answer as a simplified fraction.

A) $\frac { 25 } { 102 }$

B) $\frac { 13 } { 51 }$

C) $\frac { 25 } { 51 }$

D) $\frac { 1 } { 2,652 }$

Question 53

A die with 12 sides is rolled. What is the probability of rolling a number less than 11?

A) $\frac { 1 } { 12 }$

B) $\frac { 5 } { 6 }$

C) 10

D) $\frac { 11 } { 12 }$

Question 54

Find the indicated probability.

-If a person is randomly selected, find the probability that his or her birthday is in May. Ignore leap years.

A) $\frac { 1 } { 31 }$

B) $\frac { 1 } { 12 }$

C) $\frac { 31 } { 365 }$

D) $\frac { 1 } { 365 }$

Question 55

From the information provided, create the sample space of possible outcomes.

-Flip a coin twice.

A) HH HT TT
B) HT TH
C) HH TT HT HT
D) HH HT TH TT

Question 56

Find the probability of correctly answering the first 5 questions on a multiple choice test if random guesses are made and each question has 4 possible answers.

A) $\frac { 4 } { 5 }$

B) $\frac { 1 } { 1024 }$

C) $\frac { 1 } { 625 }$

D) $\frac { 5 } { 4 }$

Question 57

A study conducted at a certain college shows that 57% of the school's graduates find a job in their chosen field within a year after graduation. Find the probability that 9 randomly selected graduates all find jobs in their chosen field within a year of graduating. Round to the nearest thousandth if necessary.

A) 5.130
B) 0.006
C) 0.158
D) 0.011

Question 58

Evaluate the expression.

- ${ } _ { 8 } \mathrm { P } _ { 4 }$

A) 70
B) 2
C) 4
D) 1680

Question 59

Find the odds against correctly guessing the answer to a multiple choice question with 3 possible answers.

A) 3 : 1
B) 2 : 1
C) 3 : 2
D) 2 : 3

Question 60

There are 9 members on a board of directors. If they must elect a chairperson, a secretary, and a treasurer, how many different slates of candidates are possible?

A) 362,880
B) 729
C) 84
D) 504

Question 61

There are 10 members on a board of directors. If they must form a subcommittee of 4 members, how many different subcommittees are possible?

A) 10,000
B) 210
C) 24
D) 5040

Question 62

Find the indicated complement.

-If $P ( A ) = \frac { 2 } { 19 }$ , find $P ( \bar { A } )$

A) 0

B) $\frac { 19 } { 2 }$

C) $\frac { 2 } { 21 }$

D) $\frac { 17 } { 19 }$

Question 63

From the information provided, create the sample space of possible outcomes.

-Two white mice mate. The male has both a white and a black fur-color gene. The female has only white fur-color genes. The fur color of the offspring depends on the pairs of fur-color genes that they receive. Assume that neither the white nor the black gene dominates. List the possible
Outcomes.

A) WW, WW
B) WW, BW
C) WB, BW
D) WW, BB

Question 64

The table below describes the smoking habits of a group of asthma sufferers.

$\begin{array}{r|ccccr} &&\text { Occasional }&\text {Regular}&\text { Heavy }\\& \text { Nonsmoker } & \text { smoker } & \text { smoker } & \text { smoker } & \text { Total } \\\hline \text { Men } & 367 & 50 & 65 & 47 & 529 \\\text { Women } & 316 & 31 & 70 & 45 & 462 \\\text { Total } & 683 & 81 & 135 & 92 & 991\end{array}$
If one of the 991 people is randomly selected, find the probability of getting a regular or heavy smoker.

A) 0.229
B) 0.493
C) 0.136
D) 0.113

Question 65

Evaluate the expression

- ${ } _ { 10 } \mathrm { P } _ { 3 }$

A) 27
B) 720
C) 7
D) 120

Question 66

In a certain town, 25% of people commute to work by bicycle. If a person is selected randomly from the town, what are the odds against selecting someone who commutes by bicycle?

A) 1 : 3
B) 1 : 4
C) 3 : 4
D) 3 : 1

Question 67

Find the indicated probability.

-You are dealt two cards successively (without replacement) from a shuffled deck of 52 playing cards. Find the probability that the first card is a King and the second card is a queen. Express your answer as a simplified fraction.

A) $\frac { 2 } { 13 }$

B) $\frac { 13 } { 102 }$

C) $\frac { 4 } { 663 }$

D) $\frac { 1 } { 663 }$

Question 68

Is Event B dependent or independent of Event A?

-A: A green ball is drawn from a box with five balls and placed next to the box. B: A red ball is drawn next and placed next to the green one.

A) Dependent
B) Independent

Question 69

From the information provided, create the sample space of possible outcomes.

-Friskie is having her fifth litter. The prior litters have either been three normal pups or two normal pups and a runt. Assume the probability of either outcome is 50%.

A) N NN NR NNN NRN
B) NR NNR NNR
C) NNR NNN
D) NNN RNN NR

Question 70

Find the indicated probability. Express your answer as a simplified fraction unless otherwise noted.

-The table below shows the soft drinks preferences of people in three age groups. $$\begin{array} { r | c c c } & \text { cola } & \text { root beer } & \text { lemon-lime } \\\hline \text { under 21 years of age } & 40 & 25 & 20 \\\text { between 21 and 40 } & 35 & 20 & 30 \\\text { over 40 years of age } & 20 & 30 & 35\end{array}$$
If one of the 255 subjects is randomly selected, find the probability that the person is over 40 and drinks cola.

A) $\frac { 4 } { 51 }$

B) $\frac { 4 } { 17 }$

C) $\frac { 4 } { 19 }$

D) None of the above is correct.

Question 71

Is Event B dependent or independent of Event A?

-A: A mosquito lands on your arm.
B: You get a mosquito bite.

A) Independent
B) Dependent

Question 72

Find the indicated probability.

-When a pair of dice are rolled there are 36 different possible outcomes: 1-1, 1-2, ... 6-6. If a pair of dice are rolled 3 times, what is the probability of getting a sum of 5 every time? Round to eight decimal places.

A) 0.00267918
B) 0.00137174
C) 0.0667
D) 0.008

Question 73

Use Bayes' theorem to find the indicated probability.

-Some employers use lie detector tests to screen job applicants. Lie detector tests are not completely reliable. Suppose that in a lie detector test, 63% of lies are identified as lies and that 16% of true statements are also identified as lies.
A company gives its job applicants a polygraph test, asking "Did you tell the truth on your job application?". Suppose that 90% of the job applicants tell the truth during the polygraph test.
What is the probability that a person who fails the test was actually telling the truth?

A) 0.438
B) 0.696
C) 0.304
D) 0.16

Question 74

Find the indicated probability. Round to the nearest thousandth.
A sample of 4 different calculators is randomly selected from a group containing 14 that are defective and 39 that have no defects. What is the probability that at least one of the calculators is
Defective?

A) 0.281
B) 0.719
C) 0.707
D) 0.140

Question 75

Answer the question, considering an event to be "unusual" if its probability is less than or equal to 0.05.
If you are told that a mystery person's name begins with a consonant, would it be "unusual" to guess the first letter of that person's name?

A) Yes
B) No

Question 76

Express the indicated degree of likelihood as a probability value.
"It will definitely turn dark tonight."

A) 1
B) 0.5
C) 0.67
D) 0.30

Question 77

The table below describes the smoking habits of a group of asthma sufferers.
$\begin{array}{r|cccr} & &\text { Light } & \text { Heavy }\\& \text { Nonsmoker } & \text { smoker } & \text { smoker } & \text { Total } \\\hline \text { Men } & 404 & 34 & 32 & 470 \\\text { Women } & 375 & 39 & 50 & 464 \\\text { Total } & 779 & 73 & 82 & 934\end{array}$

If two different people are randomly selected from the 934 subjects, find the probability that they are both women. Round to four decimal places.

A) 0.2465
B) 0.000004645
C) 0.2468
D) 0.1612

Question 78

Find the probability that 3 randomly selected people all have the same birthday. Ignore leap years. Round to eight decimal places.

A) 0.3333
B) 0.00000002
C) 0.0082
D) 0.00000751

Question 79

Find the indicated probability.

-A batch consists of 12 defective coils and 88 good ones. Find the probability of getting two good coils when two coils are randomly selected if the first selection is replaced before the second is made.

A) 0.0144
B) 0.176
C) 0.7744
D) 0.7733

Question 80

A 6-sided die is rolled. Find P(3 or 5).

A) $\frac { 1 } { 6 }$

B) $\frac { 1 } { 3 }$

C) 2

D) $\frac { 1 } { 36 }$