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# Probability Topics: Homework

Summary: This module provides a number of homework exercises related to Probability. Note: This module is currently under revision, and its content is subject to change. This module is being prepared as part of a statistics textbook that will be available for the Fall 2008 semester.

Note: You are viewing an old version of this document. The latest version is available here.

## Exercise 1

Suppose that you have 8 cards. 5 are green and 3 are yellow. The 5 green cards are numbered 1, 2, 3, 4, and 5. The 3 yellow cards are numbered 1, 2, and 3. The cards are well shuffled. You randomly draw one card. Consider the following events:

GG = card drawn is green; EE = card drawn is even-numbered

• a: List the sample space.
• b: P(G) =P(G) =
• c: P(G|E) = P(G|E) =
• d: P(G and E) = P(G and E) =
• e: P(G or E) =P(G or E) =
• f: Are GG and EE mutually exclusive? Justify your answer numerically.

### Solution

• a: {G1, G2, G3, G4, G5, Y1, Y2, Y3}{G1, G2, G3, G4, G5, Y1, Y2, Y3}
• b: 5 8 5 8
• c: 2 3 2 3
• d: 2 8 2 8 size 12{ { { size 8{2} } over { size 8{8} } } } {}
• e: 6 8 6 8 size 12{ { { size 8{6} } over { size 8{8} } } } {}
• f: No

## Exercise 2

Refer to problem (1) above. Suppose that this time you randomly draw two cards, one at a time, and with replacement.

G 1 = G 1 = size 12{G rSub { size 8{1} } ={}} {} first card is green; G 2 = G 2 = size 12{G rSub { size 8{2} } ={}} {} second card is green.

• Draw a tree diagram of the situation.
• P ( G 1 and G 2 ) = P ( G 1 and G 2 ) = size 12{P $$G rSub { size 8{1} } " and "G rSub { size 8{2} }$$ ={}} {}
• P ( at least one green ) = P ( at least one green ) = size 12{P $$"at least one green"$$ ={}} {}
• P ( G 2 G 1 ) = P ( G 2 G 1 ) = size 12{P $$G rSub { size 8{2} } \lline G rSub { size 8{1} }$$ ={}} {}
• Are G 2 G 2 size 12{G rSub { size 8{2} } } {} and G 1 G 1 size 12{G rSub { size 8{1} } } {} independent events? Explain why or why not.

## Exercise 3

Refer to problem (1) above. Suppose that this time you randomly draw two cards, one at a time, and without replacement.

G1 G1 = first card is green; G2 G2 = second card is green.

• a: Draw a tree diagram of the situation.
• P( G1 and G2 ) = P( G1 and G2 ) =
• b: P(at least one green) =P(at least one green) =
• c: P( G2 | G1 ) =P( G2 | G1 ) =
• d: Are G2 G2 and G1 G1 independent events? Explain why or why not.

### Solution

• b: ( 5 8 ) ( 4 7 ) ( 5 8 ) ( 4 7 ) size 12{ $${ { size 8{5} } over { size 8{8} } }$$ $${ { size 8{4} } over { size 8{7} } }$$ } {}
• c: ( 5 8 ) ( 3 7 ) + ( 3 8 ) ( 5 7 ) + ( 5 8 ) ( 4 7 ) ( 5 8 ) ( 3 7 ) + ( 3 8 ) ( 5 7 ) + ( 5 8 ) ( 4 7 ) size 12{ $${ { size 8{5} } over { size 8{8} } }$$ $${ { size 8{3} } over { size 8{7} } }$$ + $${ { size 8{3} } over { size 8{8} } }$$ $${ { size 8{5} } over { size 8{7} } }$$ + $${ { size 8{5} } over { size 8{8} } }$$ $${ { size 8{4} } over { size 8{7} } }$$ } {}
• d: 4 7 4 7 size 12{ { { size 8{4} } over { size 8{7} } } } {}
• e: No

## Exercise 4

Roll two fair dice. Each die has 6 faces.

• a: Let AA be the event that either AA 3 or 4 is rolled first, followed by an even number. Find P(A)P(A).
• b: Let BB be the event that the sum of the two rolls is at most 7. Find P(B)P(B).
• c: In words, explain what “P(A|B)P(A|B)” represents. Find P(A|B)P(A|B).
• d: Are AA and BB mutually exclusive events? Explain your answer in 1 - 3 complete sentences, including numerical justification.
• e: Are AA and BB independent events? Explain your answer in 1 - 3 complete sentences, including numerical justification.

## Exercise 5

A special deck of cards has 10 cards. Four are green, three are blue, and three are red. When a card is picked, the color of it is recorded. An experiment consists of first picking a card and then tossing a coin.

• a: List the sample space.
• b: Let AA be the event that a blue card is picked first, followed by landing a head on the coin toss. Find P(A)P(A).
• c: Let BB be the event that a red or green is picked, followed by landing a head on the coin toss. Are the events AA and BB mutually exclusive? Explain your answer in 1 - 3 complete sentences, including numerical justification.
• d: Let CC be the event that a red or blue is picked, followed by landing a head on the coin toss. Are the events AA and CC mutually exclusive? Explain your answer in 1 - 3 complete sentences, including numerical justification.

### Solution

• a: { GH , GT , BH , BT , RH , RT } { GH , GT , BH , BT , RH , RT } size 12{ lbrace ital "GH", ital "GT", ital "BH", ital "BT", ital "RH", ital "RT" rbrace } {}
• b: 3 20 3 20 size 12{ { { size 8{3} } over { size 8{"20"} } } } {}
• c: Yes
• Nod:

## Exercise 6

An experiment consists of first rolling a die and then tossing a coin:

• a: List the sample space.
• b: Let AA be the event that either a 3 or 4 is rolled first, followed by landing a head on the coin toss. Find P(A)P(A).
• c: Let BB be the event that a number less than 2 is rolled, followed by landing a head on the coin toss. Are the events AA and BB mutually exclusive? Explain your answer in 1 - 3 complete sentences, including numerical justification.

## Exercise 7

An experiment consists of tossing a nickel, a dime and a quarter. Of interest is the side the coin lands on.

• a: List the sample space.
• b: Let AA be the event that there are at least two tails. Find P(A)P(A).
• c: Let BB be the event that the first and second tosses land on heads. Are the events AA and BB mutually exclusive? Explain your answer in 1 - 3 complete sentences, including justification.

### Solution

• a: { ( HHH ) , ( HHT ) , ( HTH ) , ( HTT ) , ( THH ) , ( THT ) , ( TTH ) , ( TTT ) } { ( HHH ) , ( HHT ) , ( HTH ) , ( HTT ) , ( THH ) , ( THT ) , ( TTH ) , ( TTT ) } size 12{ lbrace $$ital "HHH"$$ , $$ital "HHT"$$ , $$ital "HTH"$$ , $$ital "HTT"$$ , $$ital "THH"$$ , $$ital "THT"$$ , $$ital "TTH"$$ , $$ital "TTT"$$ rbrace } {}
• b: 4 8 4 8 size 12{ { { size 8{4} } over { size 8{8} } } } {}
• c: Yes

## Exercise 8

Let P(C) = 0.4; P(D) = 0.5; P(C|D) = 0.6P(C)=0.4; P(D)=0.5; P(C|D)=0.6 .

• a: Find P(C and D)P(C and D) .
• b: Are CC and DD mutually exclusive? Why or why not?
• c: Are CC and DD independent events? Why or why not?
• d: Find P(C and D)P(C and D) .
• e: Find P(D|C)P(D|C).

## Exercise 9

EE size 12{E} {} and FF size 12{F} {} mutually exclusive events. P(E)=0.4P(E)=0.4 size 12{P $$E$$ =0 "." 4} {}; P(F)=0.5P(F)=0.5 size 12{P $$F$$ =0 "." 5} {}. Find P(EF)P(EF) size 12{P $$E \lline F$$ } {}.

Rewrite the basic Addition Rule (P(Y(P(Y size 12{ $$P \( Y} {}or Z)=P(Y)+P(Z)P(YZ)=P(Y)+P(Z)P(Y size 12{Z$$ =P $$Y$$ +P $$Z$$ - P $$Y} {}and Z))Z)) size 12{Z$$ \) } {} using the information that YY size 12{Y} {} and ZZ size 12{Z} {} are independent events. Use the rewritten rule to find P(Z)P(Z) size 12{P $$Z$$ } {}if P(YP(Y size 12{P $$Y} {}or Z)=0.71Z)=0.71 size 12{Z$$ =0 "." "71"} {}and P(Y)=0.42P(Y)=0.42 size 12{P $$Y$$ =0 "." "42"} {}.

0

## Exercise 10

JJ size 12{J} {} and KK size 12{K} {} are independent events. P ( J K ) = 0 . 3 P ( J K ) = 0 . 3 size 12{P $$J \lline K$$ =0 "." 3} {} Find P(J)P(J) size 12{P $$J$$ } {} .

## Exercise 11

UU size 12{U} {} and VV size 12{V} {} are mutually exclusive events. P(U)=0.26P(U)=0.26 size 12{P $$U$$ =0 "." "26"} {}; P(V)=0.37P(V)=0.37 size 12{P $$V$$ =0 "." "37"} {}. Find:

• a: P(UP(U size 12{P $$U} {} and V)V) size 12{V$$ } {}
• b: P ( U V ) P ( U V ) size 12{P $$U \lline V$$ } {}
• c: P(UP(U size 12{P $$U} {} or V)V) size 12{V$$ } {}

• a: 0
• b: 0
• c: 0.63

## Exercise 12

QQ size 12{Q} {}and RR size 12{R} {} are independent events. P(Q)=0.4P(Q)=0.4 size 12{P $$Q$$ =0 "." 4} {}; P(QP(Q size 12{P $$Q} {} and R)=0.10R)=0.10 size 12{R$$ =0 "." "10"} {} . Find P(R)P(R) size 12{P $$R$$ } {}.

## Exercise 13

YY size 12{Y} {} and ZZ size 12{Z} {} are independent events:

• a: Rewrite the basic Addition Rule ( P(Y or Z) = P(Y) + P(Z) - P(Y and Z) ) using the information that Y and Z are independent events.
• b: Use the rewritten rule to find P(Z) if P(Y or Z) = 0.71 and P(Y) = 0.42 .

• b: 0.5

## Exercise 14

GG size 12{G} {} and HH size 12{H} {} are mutually exclusive events. P(G)=0.5P(G)=0.5 size 12{P $$G$$ =0 "." 5} {}; P(H)=0.3P(H)=0.3 size 12{P $$H$$ =0 "." 3} {}

• a: Explain why the following statement MUST be false: P ( H G ) = 0 . 4 P ( H G ) = 0 . 4 size 12{P $$H \lline G$$ =0 "." 4} {} .
• b: Find: P(HP(H size 12{P $$H} {} or G)G) size 12{G$$ } {}.
• c: Are GG size 12{G} {} and HH size 12{H} {} independent or dependent events? Explain in a complete sentence.

## Exercise 15

The following are real data from Santa Clara County, CA. As of March 31, 2000, there was a total of 3059 documented cases of AIDS in the county. They were grouped into the following categories (Source: Santa Clara County Public H.D.):

Table 1: * includes homosexual/bisexual IV drug users
Homosexual/Bisexual IV Drug User* Heterosexual Contact Other
female 0 70 136 49
male 2146 463 60 135

Suppose one of the persons with AIDS in Santa Clara County is randomly selected. Compute the following:

• a: P(P( size 12{P $$} {} person is female )=______)=______ size 12{$$ ="______"} {}
• b: P(P( size 12{P $$} {} person has a risk factor heterosexual contact )=______)=______ size 12{$$ ="______"} {}
• c: P(P( size 12{P $$} {}person is female OR has a risk factor of IV Drug User )=______)=______ size 12{$$ ="______"} {}
• d: P(P( size 12{P $$} {}person is female AND has a risk factor of homosexual/bisexual )=______)=______ size 12{$$ ="______"} {}
• e: P(P( size 12{P $$} {}person is male AND has a risk factor of IV Drug User )=______)=______ size 12{$$ ="______"} {}
• f: P(P( size 12{P $$} {}female GIVEN person got the disease from heterosexual contact )=______)=______ size 12{$$ ="______"} {}
• g: Construct a Venn Diagram. Make one group females and the other group heterosexual contact.

### Solution

Column totals are: 2146, 533, 196, 174; Row totals are: 255, 2804; Total in the survey: 3059

• a: 255 3059 255 3059
• b: 196 3059 196 3059
• c: 718 3059 718 3059 size 12{ { { size 8{"718"} } over { size 8{"3059"} } } } {}
• d: 0
• e: 463 3059 463 3059
• f: 136 196 136 196

## Exercise 16

Solve these questions using probability rules. Do NOT use the contingency table above. 3059 cases of AIDS had been reported in Santa Clara County, CA, through March 31, 2000. Those cases will be our population. Of those cases, 6.4% obtained the disease through heterosexual contact and 7.4% are female. Out of the females with the disease, 53.3% got the disease from heterosexual contact.

• a: P(person is female) = P(person is female) =
• b: P(person obtained the disease through heterosexual contact) = P(person obtained the disease through heterosexual contact) =
• c: P(female GIVEN person got the disease from heterosexual contact) =P(female GIVEN person got the disease from heterosexual contact) =
• d: Construct a Venn Diagram. Make one group females and the other group heterosexual contact. Fill in all values as probabilities.

## Exercise 17

The following table identifies a group of children by one of four hair colors, and by type of hair.

Table 2
Hair Type Brown Blond Black Red Totals
Wavy 20   15 3 43
Straight 80 15   12
Totals   20     215
• a: Complete the table above.
• b: What is the probability that a randomly selected child will have wavy hair?
• c: What is the probability that a randomly selected child will have either brown or blond hair?
• d: What is the probability that a randomly selected child will have wavy brown hair?
• e: What is the probability that a randomly selected child will have red hair, given that he has straight hair?
• f: If B is the event of a child having brown hair, find the probability of the complement of B.
• g: In words, what does the complement of B represent?

### Solution

• b: 4321543215
• c: 120 215 120 215 size 12{ { { size 8{"120"} } over { size 8{"215"} } } } {}
• d: 2021520215
• e: 1217212172
• f: 115 215 115 215 size 12{ { { size 8{"115"} } over { size 8{"215"} } } } {}

## Exercise 18

A previous year, the weights of the members of the San Francisco 49ers and the Dallas Cowboys were published in the San Jose Mercury News. The factual data are compiled into the following table.

Table 3
Shirt# ≤ 210 211-250 251-290 290≤
1-33 21 5 0 0
34-66 6 18 7 4
66-99 6 12 22 5

### For the following, suppose that you randomly select one player from the 49ers or Cowboys.

• a: Find the probability that his shirt number is from 1 to 33.
• b: Find the probability that he weighs at most 210 pounds.
• c: Find the probability that his shirt number is from 1 to 33 AND he weighs at most 210 pounds.
• d: Find the probability that his shirt number is from 1 to 33 OR he weighs at most 210 pounds.
• e: Find the probability that his shirt number is from 1 to 33 GIVEN that he weighs at most 210 pounds.
• f: If having a shirt number from 1 to 33 and weighing at most 210 pounds were independent events, then what should be true about P(P( size 12{P $$} {}shirt 133210133210 size 12{#1 - "33" \lline <= "210"} {} pounds )) size 12{$$ } {}?

## Exercise 19

Approximately 249,000,000 people live in the United States. Of these people, 31,800,000 speak a language other than English at home. Of those who speak another language at home, over 50 percent speak Spanish. (Source: U.S. Bureau of the Census, 1990 Census)

Let: EE = speak English at home; E'E' = speak another language at home; SS = speak Spanish at home

 a. P(E') = i. 0.8723 b. P(E) = ii. > 0.50 c. P(S) = iii. 0.1277 d. P(S|E') = iv. > 0.0639

• a: iii
• b: i
• c: iv
• d: ii

## Exercise 20

The probability that a male develops some form of cancer in his lifetime is 0.4567 (Source: American Cancer Society). The probability that a male has at least one false positive test result (meaning the test comes back for cancer when the man does not have it) is 0.51 (Source: USA Today). Some of the questions below do not have enough information for you to answer them. Write “not enough information” for those answers.

Let: CC = a man develops cancer in his lifetime; P P = man has at least one false positive

• a: Construct a tree diagram of the situation.
• b: P(C)P(C) = __________
• c: P(P|C)P(P|C) = __________
• d: P(P|C' )P(P|C' ) = __________
• e: If a test comes up positive, based upon numerical values, can you assume that man has cancer? Justify numerically and explain why or why not.

## Exercise 21

In 1994, the U.S. government held a lottery to issue 55,000 Green Cards (permits for non-citizens to work legally in the U.S.). Renate Deutsch, from Germany, was one of approximately 6.5 million people who entered this lottery. Let G = won Green Card

• a: What was Renate’s chance of winning a Green Card? Write your answer as a probability statement.
• b: In the summer of 1994, Renate received a letter stating she was one of 110,000 finalists chosen. Once the finalists were chosen, assuming that each finalist had an equal chance to win, what was Renate’s chance of winning a Green Card? Let FF = was a finalist. Write your answer as a conditional probability statement.
• c: Are GG and FF independent or dependent events? Justify your answer numerically and also explain why.
• d: Are GG and FF mutually exclusive events? Justify your answer numerically and also explain why.

### Note:

P.S. Amazingly, on 2/1/95, Renate learned that she would receive her Green Card -- true story!

### Solution

• a: P ( G ) = 0 . 008 P ( G ) = 0 . 008 size 12{P $$G$$ =0 "." "008"} {}
• b: 0.5
• c: dependent
• d: No

## Exercise 22

Three professors at George Washington University did an experiment to determine if economists are more selfish than other people. They dropped 64 stamped, addressed envelopes with \$10 cash in different classrooms on the George Washington campus. 44% were returned overall. From the economics classes 56% of the envelopes were returned. From the business, psychology, and history classes 31% were returned. (Source: Wall Street Journal)

Let: RR = money returned; EE = economics classes; OO = other classes

• a: Write a probability statement for the overall percent of money returned.
• b: Write a probability statement for the percent of money returned out of the economics classes.
• c: Write a probability statement for the percent of money returned out of the other classes.
• d: Is money being returned independent of the class? Justify your answer numerically and explain it.
• e: Based upon this study, do you think that economists are more selfish than other people? Explain why or why not. Include numbers to justify your answer.

## Exercise 23

Table 5
Race and Sex 1 - 14 15 - 24 25 - 64 over 64 TOTALS
white, male 210 3360 13,610   22,050
white, female 80 580 3380   4930
black, male 10 460 1060   1670
black, female 0 40 270   330
all others
TOTALS 310 4650 18,780   29,760
• a: Fill in the column for the suicides for individuals over age 64.
• b: Fill in the row for all other races.
• c: Find the probability that a randomly selected individual was a white male.
• d: Find the probability that a randomly selected individual was a black female.
• e: Find the probability that a randomly selected individual was black
• f: Comparing “Race and Sex” to “Age,” which two groups are mutually exclusive? How do you know?

### Note:

Do not include “all others” for the following:
• g: Find the probability that a randomly selected individual was male.
• h: Out of the individuals over age 64, find the probability that a randomly selected individual was a black or white male.
• i: Are being male and committing suicide over age 64 independent events? How do you know?

### Solution

• c: 22050 29760 22050 29760 size 12{ { {"22050"} over {"29760"} } } {}
• d: 330 29760 330 29760 size 12{ { {"330"} over {"29760"} } } {}
• e: 2000 29760 2000 29760 size 12{ { { size 8{"2000"} } over { size 8{"29760"} } } } {}
• f: 23720 29760 23720 29760 size 12{ { { size 8{"23720"} } over { size 8{"29760"} } } } {}
• g: 5010 6020 5010 6020 size 12{ { { size 8{"5010"} } over { size 8{"6020"} } } } {}
• h: Black females and ages 1-14
• i: No

The following refers to questions (24) and (25): The percent of licensed U.S. drivers (from a recent year) that are female is 48.60. Of the females, 5.03% are age 19 and under; 81.36% are age 20 - 64; 13.61% are age 65 or over. Of the licensed U.S. male drivers, 5.04% are age 19 and under; 81.43% are age 20 - 64; 13.53% are age 65 or over. (Source: Federal Highway Administration, U.S. Dept. of Transportation)

## Exercise 24

Complete the following:

• a: Construct a table or a tree diagram of the situation.
• P(P( size 12{P $$} {}driver is female )=)= size 12{$$ ={}} {} __________
• b: P(P( size 12{P $$} {}driver is age 65 or over | driver is female) )=)= size 12{$$ ={}} {} __________
• c: P(P( size 12{P $$} {}driver is age 65 or over AND female )=)= size 12{$$ ={}} {}__________
• d: In words, explain the difference between the probabilities in part (c) and part (d).
• e: P(P( size 12{P $$} {}driver is age 65 or over )=)= size 12{$$ ={}} {} __________
• f: Are being age 65 or over and being female independent events? How do you know?
• g: Are being age 65 or over and being female mutually exclusive events? How do you know

## Exercise 25

Suppose that 10,000 U.S. licensed drivers are randomly selected.

• a: How many would you expect to be male?
• b: Using the table or tree diagram from problem (21), construct a contingency table of gender versus age group.
• c: Using the contingency table, find the probability that out of the age 20 - 64 group, a randomly selected driver is female.

• a: 5140
• c: 0.49

## Exercise 26

Approximately 86.5% of Americans commute to work by car, truck or van. Out of that group, 84.6% drive alone and 15.4% drive in a carpool. Approximately 3.9% walk to work and approximately 5.3% take public transportation. (Source: Bureau of the Census, U.S. Dept. of Commerce. Disregard rounding approximations.)

• a: Construct a table or a tree diagram of the situation. Include a branch for all other modes of transportation to work.
• b: Assuming that the walkers walk alone, what percent of all commuters travel alone to work?
• c: Suppose that 1000 workers are randomly selected. How many would you expect to travel alone to work?
• d: Suppose that 1000 workers are randomly selected. How many would you expect to drive in a carpool?

## Exercise 27

Explain what is wrong with the following statements. Use complete sentences.

• a: If there’s a 60% chance of rain on Saturday and a 70% chance of rain on Sunday, then there’s a 130% chance of rain over the weekend.
• b: The probability that a baseball player hits a home run is greater than the probability that he gets a successful hit.

## Try these multiple choice questions.

Questions 28 – 29 refer to the following probability tree diagram which shows tossing an unfair coin FOLLOWED BY drawing one bead from a cup containing 3 red ( RR size 12{R} {}), 4 yellow ( YY size 12{Y} {}) and 5 blue ( BB size 12{B} {}) beads. For the coin, P(H)=23P(H)=23 size 12{P $$H$$ = { {2} over {3} } } {} and P(T)=13P(T)=13 size 12{P $$T$$ = { {1} over {3} } } {} where H=H= size 12{H={}} {}“heads” and T=T= size 12{T={}} {} “tails.”

### Note:

Redo this Tree in Photoshop, it's bad - Uchenna

### Exercise 28

Find P(P( size 12{P $$} {}tossing a Head on the coin AND a Red bead )) size 12{$$ } {}

• A: 2 3 2 3 size 12{ { {2} over {3} } } {}
• B: 5 15 5 15 size 12{ { {5} over {"15"} } } {}
• C: 6 36 6 36 size 12{ { {6} over {"36"} } } {}
• D: 5 36 5 36 size 12{ { {5} over {"36"} } } {}

C

### Exercise 29

• A: 15 36 15 36 size 12{ { {"15"} over {"36"} } } {}
• B: 10 36 10 36 size 12{ { {"10"} over {"36"} } } {}
• C: 10 12 10 12 size 12{ { {"10"} over {"12"} } } {}
• D: 6 32 6 32 size 12{ { {6} over {"32"} } } {}

#### Solution

A

Questions 30 – 32 refer to the following table of data obtained from www.baseball-almanac.com showing hit information for 4 well known baseball players.

Table 6
NAME Single Double Triple Home Run TOTAL HITS
Babe Ruth 1517 506 136 714 2873
Jackie Robinson 1054 273 54 137 1518
Ty Cobb 3603 174 295 114 4189
Hank Aaron 2294 624 98 755 3771
TOTAL 8471 1577 583 1720 12351

### Exercise 30

Find P(P( size 12{P $$} {}hit was made by Babe Ruth )) size 12{$$ } {}:

• A: 1518 2873 1518 2873 size 12{ { {"1518"} over {"2873"} } } {}
• B: 2873 12351 2873 12351 size 12{ { {"2873"} over {"12351"} } } {}
• C: 583 12351 583 12351 size 12{ { {"583"} over {"12351"} } } {}
• D: 4189 12351 4189 12351 size 12{ { {"4189"} over {"12351"} } } {}

B

### Exercise 31

Find P(P( size 12{P $$} {}hit was made by Ty Cobb | the hit was a Home Run )) size 12{$$ } {}

• A: 4189 12351 4189 12351 size 12{ { {"4189"} over {"12351"} } } {}
• B: 1141 1720 1141 1720 size 12{ { {"1141"} over {"1720"} } } {}
• C: 1720 4189 1720 4189 size 12{ { {"1720"} over {"4189"} } } {}
• D: 114 12351 114 12351 size 12{ { {"114"} over {"12351"} } } {}

B

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