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

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.

  • G = card drawn is green
  • E = card drawn is even-numbered
  • List the sample space.
  • P(G) =
  • P(G|E) =
  • P(G AND E) =
  • P(G OR E) =
  • Are G and E mutually exclusive? Justify your answer numerically.
  • {G1, G2, G3, G4, G5, Y1, Y2, Y3}
  • 5 8
  • 2 3
  • 2 8 size 12{ { { size 8{2} } over { size 8{8} } } } {}
  • 6 8 size 12{ { { size 8{6} } over { size 8{8} } } } {}
  • No

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

Refer to the previous problems. Suppose that this time you randomly draw two cards, one at a time, and without replacement .

  • G 1 = first card is green
  • G 2 = second card is green
  • Draw a tree diagram of the situation.
  • P( G 1  AND  G 2 ) =
  • P(at least one green) =
  • P( G 2 | G 1 ) =
  • Are G 2 and G 1 independent events? Explain why or why not.
  • ( 5 8 ) ( 4 7 ) size 12{ \( { { size 8{5} } over { size 8{8} } } \) \( { { size 8{4} } over { size 8{7} } } \) } {}
  • ( 5 8 ) ( 3 7 ) + ( 3 8 ) ( 5 7 ) + ( 5 8 ) ( 4 7 ) size 12{ \( { { size 8{5} } over { size 8{8} } } \) \( { { size 8{3} } over { size 8{7} } } \) + \( { { size 8{3} } over { size 8{8} } } \) \( { { size 8{5} } over { size 8{7} } } \) + \( { { size 8{5} } over { size 8{8} } } \) \( { { size 8{4} } over { size 8{7} } } \) } {}
  • 4 7 size 12{ { { size 8{4} } over { size 8{7} } } } {}
  • No

Roll two fair dice. Each die has 6 faces.

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

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.

  • List the sample space.
  • Let A be the event that a blue card is picked first, followed by landing a head on the coin toss. Find P(A) .
  • Let B be the event that a red or green is picked, followed by landing a head on the coin toss. Are the events A and B mutually exclusive? Explain your answer in 1 - 3 complete sentences, including numerical justification.
  • Let C be the event that a red or blue is picked, followed by landing a head on the coin toss. Are the events A and C mutually exclusive? Explain your answer in 1 - 3 complete sentences, including numerical justification.
  • { GH , GT , BH , BT , RH , RT } size 12{ lbrace ital "GH", ital "GT", ital "BH", ital "BT", ital "RH", ital "RT" rbrace } {}
  • 3 20 size 12{ { { size 8{3} } over { size 8{"20"} } } } {}
  • Yes
  • No

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

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

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

  • List the sample space.
  • Let A be the event that there are at least two tails. Find P(A) .
  • Let B be the event that the first and second tosses land on heads. Are the events A and B mutually exclusive? Explain your answer in 1 - 3 complete sentences, including justification.
  • { ( 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 } {}
  • 4 8 size 12{ { { size 8{4} } over { size 8{8} } } } {}
  • Yes

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Source:  OpenStax, Elementary statistics. OpenStax CNX. Dec 30, 2013 Download for free at http://cnx.org/content/col10966/1.4
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