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3.1 Homework (modified r. bloom)

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This module provides a number of homework exercises related to basic concepts and methods in probability. This revision of the original module by Dr. B. Illowsky and S. Dean in the textbook collection Collaborative Statistics has new problems added at the end of the module.

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.
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Source:  OpenStax, Collaborative statistics homework book: custom version modified by r. bloom. OpenStax CNX. Dec 23, 2009 Download for free at http://legacy.cnx.org/content/col10619/1.2
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