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Probability Topics: Homework

Module by: Dr. Barbara Illowsky, Susan Dean

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 1

  • 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 3

  • 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 5

  • 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 7

  • 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 9

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 11

  • 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 13

  • 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.):

* 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) =
  • g. Construct a Venn Diagram. Make one group females and the other group heterosexual contact.

Solution 15

The completed contingency table is as follows:

* 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

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.

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 1