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Homework February 8, 2005

 

 Remember, combin(x,y) = x!/((y!*(x-y)!)

 

Your top drawer contains 20 white socks and 10 black socks.
  1. You draw two socks at random without replacement. What is the probability they are the same color?

    2. Winning combinations: combin(20,2)+combin(10,2) = 190 + 45 = 235

    Total combinations: combin(30,2) = 435

    Probability = 235/435 = 54.02%.

  2. You draw four socks at random without replacement. What is the probability you draw two white socks and two black socks?

    3. Winning combinations: combin(20,2)*combin(10,2) = 190*45 = 8550

    Total combinations: combin(30,4) = 27405

    Probability: 8550/27405 = 31.20%.

  3. You draw all the socks one at a time without replacement. What is the probability the first and last socks are the same color?

    4. Same answer as problem one. The second sock and the last sock have the same odds.

    Your middle drawer contains 6 white socks, 6 black socks, 6 orange socks, 6 green socks, and 6 purple socks.
  4. You draw two socks at random without replacement. What is the probability they are the same color?

    5. Winning combinations: 5*combin(6,2) = 5*15 = 75

    Total combinations: combin(30,2) = 435

    Probability: 75/435 = 5/29 = 17.24%.

    Alternatively, once you pick the first sock there are 5 left of the same color and 29 total left, so the probability is 5/29.

  5. You draw four socks at random without replacement. What is the probability they are all different colors?

    6. There are combin(5,4)=5 ways to pick 4 colors out of 5. Then there are 6 ways to pick one sock for each of the 4 colors selected.

    Winning combinations: 5*64 = 5*1296 = 6480

    Total combinations = combin(30,4) = 27,405

    Probability = 6480/27405 = 23.65%. 

  6. You draw four socks at random without replacement. What is the probability that exactly two colors are represented?

    7. One way for two colors to be represented is for each color to have two socks. There are combin(5,2)=10 ways to choose 2 colors out of 5. Then there are combin(6,2)=15 to draw 2 socks out of 6. So there are 10*152 = 2250 ways to draw two sets of different colors composed to two socks each.

    The other way for two colors to be represented is for one color to have three socks and the other to have one. For the color with 3 socks there are 5 colors to choose from. For the color with one sock there are 4 colors left. Note that you don’t do combin(5,2) because the sets are different sizes. So there are 5*4=20 ways to choose the colors. Then there are combin(6,3)=20 ways to choose 3 socks out of 6 for the set with three socks. There are obviously six ways to pick one sock out of six for the set with one sock. So the total number of combinations for 3 socks of one color and 1 of another is 20*20*6 = 2400.

    There are combin(30,4) = 27,405 ways to draw any four socks out of 30.

    So the probability that two colors are represented out of four socks is (2250+2400)/27405 = 4650/27405 = 16.97%.

     

  7. You draw one set of four socks and one set of two socks at random and without replacement. What is the probability both sets are all of the same color and each set is a different color?

    8. Winning combinations: there are 5 ways to pick the 4-ball color and 4 left for the 2-ball color (note you don’t do combin(5,2) because the sets are different sizes. Once the colors are assigned the number of ways to pick 4 balls out of 6 for the 4-ball set is combin(6,4) = 15. The number of ways to pick 2 balls out of 6 for the 2-ball set is combin(6,2)=15. So the total number of winning combinations is 5*4*15*15=4500.

    Total combinations: There are 30!/(4!*2!*24!) = 8,906,625 ways to draw sets of 4, 2, and 24 (remaining) out of 30. Alternatively you could use combin(30,4)*combin(26,2).

    Probability = 4500/8906625 = 0.0005052.

    Your bottom drawer contains 18 black socks, 18 red socks, and 2 green socks.
  8. You draw two socks at random and without replacement. What is the probability they are both the same color?

    9. Winning combinations: combin(18,2)+combin(18,2)*combin(2,2) = 153+153*1 = 307.

    Total combinations: combin(38,2) = 703.

    Probability: 307/703 = 43.67%.

  9. Your friend offers to pay you 1 to 1 if you achieve the goal of problem 8? What is the expected value (to you) of this bet?

    10. Prob win * odds – prob loss = (307/703)*1 – (396/703) = -89/703 = -12.66%.

  10. You draw three socks at random. What is the probability they are all different colors?

    11. Winning combinations: 18*18*2=648.

    Total combinations: combin(38,3) = 8,436.

    Probability: 648/8436 = 7.68%.

    Alternatively the probability of picking black first, then red, and then green is (18/38)*(18/37)*(2/36)=1.28%. However there are 3!=6 ways to order the three colors, so multiply by 6 for the other orders of picking the colors: 6*1.28% = 7.68%.

  11. Your older sister offers 12 to 1 odds that you can achieve the goal of problem 10. What is the house (or sister’s) advantage of this bet?

Expected value = (648/8436)*12 – (1-(648/8436)) = 0.9986 – 0.9232 = -0.00142 = -0.142%.

So the house edge is +0.142%.

 

Extra Credit: Your brother has two sock drawers. One has one black sock and one white sock. The other has two black socks. However you don’t know which drawer has which socks. You open a drawer at random and pick a sock at random. You see the sock is black. What is the probability the other sock in the same drawer is also black?

The general formula for conditional probability problems is Pr(A given B) = Pr(A and B)/Pr(B). In this case Pr(second sock is black given first sock is black) = Pr(both socks are black) / Pr(first sock is black) = 0.5 / 0.75 = 2/3.