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Exercises

Module by: Kathy Chu. E-mail the author

Based on: Probability Topics: Homework by Susan Dean, Barbara Illowsky, Ph.D.

Summary: This module provides a number of homework exercises related to Probability.

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.

  • 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 the previous problem. Suppose that this time you randomly draw two cards, one at a time, and with replacement.

  • G 1 = first card is green G 1 = first card is green
  • G 2 = second card is green G 2 = second card is green
  • a. Draw a tree diagram of the situation.
  • b. P ( G 1  AND  G 2 ) = P ( G 1  AND  G 2 ) = size 12{P \( G rSub { size 8{1} } " and "G rSub { size 8{2} } \) ={}} {}
  • c. P ( at least one green ) = P ( at least one green ) = size 12{P \( "at least one green" \) ={}} {}
  • d. P ( G 2 ∣ G 1 ) = P ( G 2 ∣ G 1 ) = size 12{P \( G rSub { size 8{2} } \lline G rSub { size 8{1} } \) ={}} {}
  • e. 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 the previous problems. 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.
  • b>. P( G1  AND  G2 ) = P( G1  AND  G2 ) =
  • c. P(at least one green) =P(at least one green) =
  • d. P( G2 | G1 ) =P( G2 | G1 ) =
  • e. 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. List the sample space.
  • b. Let AA be the event that either a 3 or 4 is rolled first, followed by an even number. Find P(A)P(A).
  • c. Let BB be the event that the sum of the two rolls is at most 7. Find P(B)P(B).
  • d. In words, explain what “P(A|B)P(A|B)” represents. Find P(A|B)P(A|B).
  • e. Are AA and BB mutually exclusive events? Explain your answer in 1 - 3 complete sentences, including numerical justification.
  • f. 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
  • d. No

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

Consider the following scenario:

  • Let P(C) = 0.4P(C)=0.4
  • Let P(D) = 0.5P(D)=0.5
  • Let P(C|D) = 0.6P(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(E∣F)P(E∣F) size 12{P \( E \lline F \) } {}.

Solution

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. 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(U AND V)P(U AND V) =
  • b. P(U | V) P(U | V) =
  • c. P(U OR V)P(U OR V) =

Solution

  • 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 ; P(Q AND R) = 0.1P(Q AND R) = 0.1 . Find P(R)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) 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)P(Z) if P(Y OR Z) = 0.71P(Y OR Z) = 0.71 and P(Y) = 0.42P(Y) = 0.42 .

Solution

  • 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(H OR G)P(H OR 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 Totals
Female 0 70 136 49 ____
Male 2146 463 60 135 ____
Totals ____ ____ ____ ____ ____

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

  • a. P(person is female) P(person is female) =
  • b. P(person has a risk factor Heterosexual Contact) P(person has a risk factor Heterosexual Contact) =
  • c. P(person is female OR has a risk factor of IV Drug User) P(person is female OR has a risk factor of IV Drug User) =
  • d. P(person is female AND has a risk factor of Homosexual/Bisexual) P(person is female AND has a risk factor of Homosexual/Bisexual) =
  • e. P(person is male AND has a risk factor of IV Drug User) P(person is male AND has a risk factor of IV Drug User) =
  • f. P(female GIVEN person got the disease from heterosexual contact) P(female GIVEN person got the disease from heterosexual contact) =

Solution

The completed contingency table is as follows:

Table 2: * includes homosexual/bisexual IV drug users
  Homosexual/Bisexual IV Drug User* Heterosexual Contact Other Totals
Female 0 70 136 49 255
Male 2146 463 60 135 2804
Totals 2146 533 196 174 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

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(Shirt# 1-33 | ≤ 210 pounds) P(Shirt# 1-33 | ≤ 210 pounds)?

Exercise 17

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

Finish each probability statement by matching the correct answer.

Table 4
Probability Statements Answers
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

Solution

  • a. iii
  • b. i
  • c. iv
  • d. ii

The next two questions refer to the following: 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)

Try these multiple choice questions.

The next three questions refer to the following table of data obtained from www.baseball-almanac.com showing hit information for 4 well known baseball players. Suppose that one hit from the table is randomly selected.

Table 5
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 18

Find P(hit was made by Babe Ruth)P(hit was made by Babe Ruth).

  • 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"} } } {}

Solution

B

Exercise 19

Find P(hit was made by Ty Cobb | The hit was a Home Run)P(hit was made by Ty Cobb | The hit was a Home Run)

  • 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"} } } {}

Solution

B

Exercise 20

Are the hit being made by Hank Aaronthe hit being made by Hank Aaron and the hit being a doublethe hit being a double independent events?

  • A. Yes, because P(hit by Hank Aaron | hit is a double) = P(hit by Hank Aaron)P(hit by Hank Aaron | hit is a double) = P(hit by Hank Aaron)
  • B. No, because P(hit by Hank Aaron | hit is a double) ≠ P(hit is a double)P(hit by Hank Aaron | hit is a double) ≠ P(hit is a double)
  • C. No, because P(hit is by Hank Aaron | hit is a double) ≠ P(hit by Hank Aaron)P(hit is by Hank Aaron | hit is a double) ≠ P(hit by Hank Aaron)
  • D. Yes, because P(hit is by Hank Aaron | hit is a double) = P(hit is a double)P(hit is by Hank Aaron | hit is a double) = P(hit is a double)

Solution

C

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