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3.E: Basic Concepts of Probability (Exercises)

These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang. Complementary General Chemistry question banks can be found for other Textmaps and can be accessed here. In addition to these publicly available questions, access to private problems bank for use in exams and homework is available to faculty only on an individual basis; please contact Delmar Larsen for an account with access permission.

 

3.1 Exercises

Basic

 

A box contains 10 white and 10 black marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, two marbles in succession and noting the color each time. (To draw “with replacement” means that the first marble is put back before the second marble is drawn.)

 

A box contains 16 white and 16 black marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, three marbles in succession and noting the color each time. (To draw “with replacement” means that each marble is put back before the next marble is drawn.)

 

A box contains 8 red, 8 yellow, and 8 green marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, two marbles in succession and noting the color each time.

 

A box contains 6 red, 6 yellow, and 6 green marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, three marbles in succession and noting the color each time.

 

In the situation of Exercise 1, list the outcomes that comprise each of the following events.

At least one marble of each color is drawn.

No white marble is drawn.

 

In the situation of Exercise 2, list the outcomes that comprise each of the following events.

At least one marble of each color is drawn.

No white marble is drawn.

More black than white marbles are drawn.

 

In the situation of Exercise 3, list the outcomes that comprise each of the following events.

No yellow marble is drawn.

The two marbles drawn have the same color.

At least one marble of each color is drawn.

 

In the situation of Exercise 4, list the outcomes that comprise each of the following events.

No yellow marble is drawn.

The three marbles drawn have the same color.

At least one marble of each color is drawn.

 

Assuming that each outcome is equally likely, find the probability of each event in Exercise 5.

 

Assuming that each outcome is equally likely, find the probability of each event in Exercise 6.

 

Assuming that each outcome is equally likely, find the probability of each event in Exercise 7.

 

Assuming that each outcome is equally likely, find the probability of each event in Exercise 8.

 

A sample space is S={a,b,c,d,e}. Identify two events as U={a,b,d} and V={b,c,d}. Suppose P(a) and P(b) are each 0.2 and P(c) and P(d) are each 0.1.

Determine what P(e) must be.

Find P(U).

Find P(V).

 

A sample space is S={u,v,w,x}. Identify two events as A={v,w} and B={u,w,x}. Suppose P(u)=0.22, P(w)=0.36, and P(x)=0.27.

Determine what P(v) must be.

Find P(A).

Find P(B).

 

A sample space is S={m,n,q,r,s}. Identify two events as U={m,q,s} and V={n,q,r}. The probabilities of some of the outcomes are given by the following table:

OutcomeProbablitym0.18n0.16qr0.24s0.21

Determine what P(q) must be.

Find P(U).

Find P(V).

 

A sample space is S={d,e,f,g,h}. Identify two events as M={e,f,g,h} and N={d,g}. The probabilities of some of the outcomes are given by the following table:

OutcomeProbablityd0.22e0.13f0.27gh0.19

Determine what P(g) must be.

Find P(M).

Find P(N).

Applications

 

The sample space that describes all three-child families according to the genders of the children with respect to birth order was constructed in Note 3.9 "Example 4". Identify the outcomes that comprise each of the following events in the experiment of selecting a three-child family at random.

At least one child is a girl.

At most one child is a girl.

All of the children are girls.

Exactly two of the children are girls.

The first born is a girl.

 

The sample space that describes three tosses of a coin is the same as the one constructed in Note 3.9 "Example 4" with “boy” replaced by “heads” and “girl” replaced by “tails.” Identify the outcomes that comprise each of the following events in the experiment of tossing a coin three times.

The coin lands heads more often than tails.

The coin lands heads the same number of times as it lands tails.

The coin lands heads at least twice.

The coin lands heads on the last toss.

 

Assuming that the outcomes are equally likely, find the probability of each event in Exercise 17.

 

Assuming that the outcomes are equally likely, find the probability of each event in Exercise 18.

Additional Exercises

 

The following two-way contingency table gives the breakdown of the population in a particular locale according to age and tobacco usage:

Age Tobacco Use
Smoker Non-smoker
Under 30 0.05 0.20
Over 30 0.20 0.55

A person is selected at random. Find the probability of each of the following events.

The person is a smoker.

The person is under 30.

The person is a smoker who is under 30.

 

The following two-way contingency table gives the breakdown of the population in a particular locale according to party affiliation (A, B, C, or None) and opinion on a bond issue:

Affiliation Opinion
Favors Opposes Undecided
A 0.12 0.09 0.07
B 0.16 0.12 0.14
C 0.04 0.03 0.06
None 0.08 0.06 0.03

A person is selected at random. Find the probability of each of the following events.

The person is affiliated with party B.

The person is affiliated with some party.

The person is in favor of the bond issue.

The person has no party affiliation and is undecided about the bond issue.

 

The following two-way contingency table gives the breakdown of the population of married or previously married women beyond child-bearing age in a particular locale according to age at first marriage and number of children:

Age Number of Children
0 1 or 2 3 or More
Under 20 0.02 0.14 0.08
20–29 0.07 0.37 0.11
30 and above 0.10 0.10 0.01

A woman is selected at random. Find the probability of each of the following events.

The woman was in her twenties at her first marriage.

The woman was 20 or older at her first marriage.

The woman had no children.

The woman was in her twenties at her first marriage and had at least three children.

 

The following two-way contingency table gives the breakdown of the population of adults in a particular locale according to highest level of education and whether or not the individual regularly takes dietary supplements:

Education Use of Supplements
Takes Does Not Take
No High School Diploma 0.04 0.06
High School Diploma 0.06 0.44
Undergraduate Degree 0.09 0.28
Graduate Degree 0.01 0.02

An adult is selected at random. Find the probability of each of the following events.

The person has a high school diploma and takes dietary supplements regularly.

The person has an undergraduate degree and takes dietary supplements regularly.

The person takes dietary supplements regularly.

The person does not take dietary supplements regularly.

Large Data Set Exercises

 

Large Data Sets 4 and 4A record the results of 500 tosses of a coin. Find the relative frequency of each outcome 1, 2, 3, 4, 5, and 6. Does the coin appear to be “balanced” or “fair”?

http://www.gone.2012books.lardbucket.org/sites/all/files/data4.xls

http://www.gone.2012books.lardbucket.org/sites/all/files/data4A.xls

 

Large Data Sets 6, 6A, and 6B record results of a random survey of 200 voters in each of two regions, in which they were asked to express whether they prefer Candidate A for a U.S. Senate seat or prefer some other candidate.

Find the probability that a randomly selected voter among these 400 prefers Candidate A.

Find the probability that a randomly selected voter among the 200 who live in Region 1 prefers Candidate A (separately recorded in Large Data Set 6A).

Find the probability that a randomly selected voter among the 200 who live in Region 2 prefers Candidate A (separately recorded in Large Data Set 6B).

http://www.gone.2012books.lardbucket.org/sites/all/files/data6.xls

http://www.gone.2012books.lardbucket.org/sites/all/files/data6A.xls

http://www.gone.2012books.lardbucket.org/sites/all/files/data6B.xls

Answers

 

S={bb,bw,wb,ww}

 

 

 

S={rr,ry,rg,yr,yy,yg,gr,gy,gg}

 

 

 

{bw,wb}

{bb}

 

 

 

{rr,rg,gr,gg}

{rr,yy,gg}

 

 

 

2/4

1/4

 

 

 

4/9

3/9

0

 

 

 

0.4

0.5

0.4

 

 

 

0.21

0.6

0.61

 

 

 

{bbg,bgb,bgg,gbb,gbg,ggb,ggg}

{bbb,bbg,bgb,gbb}

{ggg}

{bgg,gbg,ggb}

{gbb,gbg,ggb,ggg}

 

 

 

7/8

4/8

1/8

3/8

4/8

 

 

 

0.25

0.25

0.05

 

 

 

0.55

0.76

0.19

0.11

 

 

 

The relative frequencies for 1 through 6 are 0.16, 0.194, 0.162, 0.164, 0.154 and 0.166. It would appear that the die is not balanced.

 

3.2: Complements, Intersections and Unions

 

Basic Exercises

 

For the sample space S={a,b,c,d,e} identify the complement of each event given.

A={a,d,e}

B={b,c,d,e}

S

 

 

For the sample space S={r,s,t,u,v} identify the complement of each event given.

R={t,u}

T={r}

∅ (the “empty” set that has no elements)

 

 

The sample space for three tosses of a coin is

S={hhh,hht,hth,htt,thh,tht,tth,ttt}

Define events

H:at least one head is observedM:more heads than tails are observed

List the outcomes that comprise H and M.

List the outcomes that comprise H ∩ M, H ∪ M, and Hc.

Assuming all outcomes are equally likely, find P(H∩M), P(H∪M), and P(Hc).

Determine whether or not Hc and M are mutually exclusive. Explain why or why not.

 

 

For the experiment of rolling a single six-sided die once, define events

T:the number rolled is threeG:the number rolled is four or greater

List the outcomes that comprise T and G.

List the outcomes that comprise T ∩ G, T ∪ G, Tc, and (T∪G)c.

Assuming all outcomes are equally likely, find P(T∩G), P(T∪G), and P(Tc).

Determine whether or not T and G are mutually exclusive. Explain why or why not.

 

 

A special deck of 16 cards has 4 that are blue, 4 yellow, 4 green, and 4 red. The four cards of each color are numbered from one to four. A single card is drawn at random. Define events

B:the card is blueR:the card is redN:the number on the card is at most two

List the outcomes that comprise B, R, and N.

List the outcomes that comprise B ∩ R, B ∪ R, B ∩ N, R ∪ N, Bc, and (B∪R)c.

Assuming all outcomes are equally likely, find the probabilities of the events in the previous part.

Determine whether or not B and N are mutually exclusive. Explain why or why not.

 

 

In the context of the previous problem, define events

Y:the card is yellowI:the number on the card is not a oneJ:the number on the card is a two or a four

List the outcomes that comprise Y, I, and J.

List the outcomes that comprise Y ∩ I, Y ∪ J, I ∩ J, Ic, and (Y∪J)c.

Assuming all outcomes are equally likely, find the probabilities of the events in the previous part.

Determine whether or not Ic and J are mutually exclusive. Explain why or why not.

 

 

The Venn diagram provided shows a sample space and two events A and B. Suppose P(a)=0.13, P(b)=0.09, P(c)=0.27, P(d)=0.20, and P(e)=0.31. Confirm that the probabilities of the outcomes add up to 1, then compute the following probabilities.

 

P(A).

P(B).

P(Ac) two ways: (i) by finding the outcomes in Ac and adding their probabilities, and (ii) using the Probability Rule for Complements.

P(A∩B).

P(A∪B) two ways: (i) by finding the outcomes in A ∪ B and adding their probabilities, and (ii) using the Additive Rule of Probability.

 

 

The Venn diagram provided shows a sample space and two events A and B. Suppose P(a)=0.32, P(b)=0.17, P(c)=0.28, and P(d)=0.23. Confirm that the probabilities of the outcomes add up to 1, then compute the following probabilities.

3ba777e453d1118c377ee69b788d0a74.jpg

 

P(A).

P(B).

P(Ac) two ways: (i) by finding the outcomes in Ac and adding their probabilities, and (ii) using the Probability Rule for Complements.

P(A∩B).

P(A∪B) two ways: (i) by finding the outcomes in A ∪ B and adding their probabilities, and (ii) using the Additive Rule of Probability.

 

 

Confirm that the probabilities in the two-way contingency table add up to 1, then use it to find the probabilities of the events indicated.

  U V W
A 0.15 0.00 0.23
B 0.22 0.30 0.10

P(A), P(B), P(A∩B).

P(U), P(W), P(U∩W).

P(U∪W).

P(Vc).

Determine whether or not the events A and U are mutually exclusive; the events A and V.

 

 

Confirm that the probabilities in the two-way contingency table add up to 1, then use it to find the probabilities of the events indicated.

  R S T
M 0.09 0.25 0.19
N 0.31 0.16 0.00

P(R), P(S), P(R∩S).

P(M), P(N), P(M∩N).

P(R∪S).

P(Rc).

Determine whether or not the events N and S are mutually exclusive; the events N and T.

 

Applications

 

Make a statement in ordinary English that describes the complement of each event (do not simply insert the word “not”).

In the roll of a die: “five or more.”

In a roll of a die: “an even number.”

In two tosses of a coin: “at least one heads.”

In the random selection of a college student: “Not a freshman.”

 

 

Make a statement in ordinary English that describes the complement of each event (do not simply insert the word “not”).

In the roll of a die: “two or less.”

In the roll of a die: “one, three, or four.”

In two tosses of a coin: “at most one heads.”

In the random selection of a college student: “Neither a freshman nor a senior.”

 

 

The sample space that describes all three-child families according to the genders of the children with respect to birth order is

S={bbb,bbg,bgb,bgg,gbb,gbg,ggb,ggg}.

For each of the following events in the experiment of selecting a three-child family at random, state the complement of the event in the simplest possible terms, then find the outcomes that comprise the event and its complement.

At least one child is a girl.

At most one child is a girl.

All of the children are girls.

Exactly two of the children are girls.

The first born is a girl.

 

 

The sample space that describes the two-way classification of citizens according to gender and opinion on a political issue is

S={mf,ma,mn,ff,fa,fn},

where the first letter denotes gender (m: male, f: female) and the second opinion (f: for, a: against, n: neutral). For each of the following events in the experiment of selecting a citizen at random, state the complement of the event in the simplest possible terms, then find the outcomes that comprise the event and its complement.

The person is male.

The person is not in favor.

The person is either male or in favor.

The person is female and neutral.

 

 

A tourist who speaks English and German but no other language visits a region of Slovenia. If 35% of the residents speak English, 15% speak German, and 3% speak both English and German, what is the probability that the tourist will be able to talk with a randomly encountered resident of the region?

 

 

In a certain country 43% of all automobiles have airbags, 27% have anti-lock brakes, and 13% have both. What is the probability that a randomly selected vehicle will have both airbags and anti-lock brakes?

 

 

A manufacturer examines its records over the last year on a component part received from outside suppliers. The breakdown on source (supplier A, supplier B) and quality (H: high, U: usable, D: defective) is shown in the two-way contingency table.

  H U D
A 0.6937 0.0049 0.0014
B 0.2982 0.0009 0.0009

The record of a part is selected at random. Find the probability of each of the following events.

The part was defective.

The part was either of high quality or was at least usable, in two ways: (i) by adding numbers in the table, and (ii) using the answer to (a) and the Probability Rule for Complements.

The part was defective and came from supplier B.

The part was defective or came from supplier B, in two ways: by finding the cells in the table that correspond to this event and adding their probabilities, and (ii) using the Additive Rule of Probability.

 

 

Individuals with a particular medical condition were classified according to the presence (T) or absence (N) of a potential toxin in their blood and the onset of the condition (E: early, M: midrange, L: late). The breakdown according to this classification is shown in the two-way contingency table.

  E M L
T 0.012 0.124 0.013
N 0.170 0.638 0.043

One of these individuals is selected at random. Find the probability of each of the following events.

The person experienced early onset of the condition.

The onset of the condition was either midrange or late, in two ways: (i) by adding numbers in the table, and (ii) using the answer to (a) and the Probability Rule for Complements.

The toxin is present in the person’s blood.

The person experienced early onset of the condition and the toxin is present in the person’s blood.

The person experienced early onset of the condition or the toxin is present in the person’s blood, in two ways: (i) by finding the cells in the table that correspond to this event and adding their probabilities, and (ii) using the Additive Rule of Probability.

 

 

The breakdown of the students enrolled in a university course by class (F: freshman,

So

: sophomore, J: junior,

Se

: senior) and academic major (S: science, mathematics, or engineering, L: liberal arts, O: other) is shown in the two-way classification table.

Major Class
F So J Se
S 92 42 20 13
L 368 167 80 53
O 460 209 100 67

A student enrolled in the course is selected at random. Adjoin the row and column totals to the table and use the expanded table to find the probability of each of the following events.

The student is a freshman.

The student is a liberal arts major.

The student is a freshman liberal arts major.

The student is either a freshman or a liberal arts major.

The student is not a liberal arts major.

 

 

The table relates the response to a fund-raising appeal by a college to its alumni to the number of years since graduation.

Response Years Since Graduation
0–5 6–20 21–35 Over 35
Positive 120 440 210 90
None 1380 3560 3290 910

An alumnus is selected at random. Adjoin the row and column totals to the table and use the expanded table to find the probability of each of the following events.

The alumnus responded.

The alumnus did not respond.

The alumnus graduated at least 21 years ago.

The alumnus graduated at least 21 years ago and responded.

 

Additional Exercises

 

The sample space for tossing three coins isS={hhh,hht,hth,htt,thh,tht,tth,ttt}

List the outcomes that correspond to the statement “All the coins are heads.”

List the outcomes that correspond to the statement “Not all the coins are heads.”

List the outcomes that correspond to the statement “All the coins are not heads.”

 

Answers

 

{b,c}

{a}

 

 

 

 

 

H={hhh,hht,hth,htt,thh,tht,tth}, M={hhh,hht,hth,thh}

H∩M={hhh,hht,hth,thh}, H∪M=H, Hc={ttt}

P(H∩M)=4/8, P(H∪M)=7/8, P(Hc)=1/8

Mutually exclusive because they have no elements in common.

 

 

 

 

 

B={b1,b2,b3,b4}, R={r1,r2,r3,r4}, N={b1,b2,y1,y2,g1,g2,r1,r2}

B∩R=∅, B∪R={b1,b2,b3,b4,r1,r2,r3,r4}, B∩N={b1,b2}, R∪N={b1,b2,y1,y2,g1,g2,r1,r2,r3,r4}, Bc={y1,y2,y3,y4,g1,g2,g3,g4,r1,r2,r3,r4}, (B∪R)c={y1,y2,y3,y4,g1,g2,g3,g4}

P(B∩R)=0, P(B∪R)=8/16, P(B∩N)=2/16, P(R∪N)=10/16, P(Bc)=12/16, P((B∪R)c)=8/16

Not mutually exclusive because they have an element in common.

 

 

 

 

 

0.36

0.78

0.64

0.27

0.87

 

 

 

 

 

P(A)=0.38, P(B)=0.62, P(A∩B)=0

P(U)=0.37, P(W)=0.33, P(U∩W)=0

0.7

0.7

A and U are not mutually exclusive because P(A∩U) is the nonzero number 0.15. A and V are mutually exclusive because P(A∩V)=0.

 

 

 

 

 

“four or less”

“an odd number”

“no heads” or “all tails”

“a freshman”

 

 

 

 

 

 

“All the children are boys.”

Event: {bbg,bgb,bgg,gbb,gbg,ggb,ggg},

Complement: {bbb}

 

 

“At least two of the children are girls” or “There are two or three girls.”

Event: {bbb,bbg,bgb,gbb},

Complement: {bgg,gbg,ggb,ggg}

 

 

“At least one child is a boy.”

Event: {ggg},

Complement: {bbb,bbg,bgb,bgg,gbb,gbg,ggb}

 

 

“There are either no girls, exactly one girl, or three girls.”

Event: {bgg,gbg,ggb},

Complement: {bbb,bbg,bgb,gbb,ggg}

 

 

“The first born is a boy.”

Event: {gbb,gbg,ggb,ggg},

Complement: {bbb,bbg,bgb,bgg}

 

 

 

 

 

 

0.47

 

 

 

 

 

0.0023

0.9977

0.0009

0.3014

 

 

 

 

 

920/1671

668/1671

368/1671

1220/1671

1003/1671

 

 

 

 

{hhh}

{hht,hth,htt,thh,tht,tth,ttt}