GAM470 Name_____Answers________

March 15, 2005 Homework

 

Questions 1-4: Consider a 5-deck blackjack game based on the playerís first two cards.

  1. How many ways can the player get a suited jack and king?
  2. There are 4 suits. Once a suit is chosen there are 5 ways to pick the jack and 5 ways to pick the king.

    4*52 = 100

  3. How many ways can the player get a suited pair of jacks, queens, or kings?
  4. There are 4 suits and 3 ranks. One the suit and rank are chosen there are combin(5,3)=10 ways to pick 2 of them out of the 5 in the shoe.

    4*3*10 = 120

  5. How many ways can the player get any other two suited cards, other than the two types in questions 1 and 2?
  6. One way to get two suited cards is two of the same card. There are 52 different cards in a deck. However 12 of these are face cards, which would result in a suited pair if the player got two. Once a specific card has been chosen there are combin(5,2)=10 ways to pick 2 of them from the 5 in the shoe. So there are (52-12)*10 = 400 ways to get 2 of the same card.

    Another ways is two different ranks. There are 4 suits. There are combin(13,2) = 78 ways to pick 2 ranks out of 13. However one of those ways results in a jack and king, which would result in a suited jack and king. Once two particular cards are chosen there are 5 of each in the deck to choose from. So there are 4*(78-1)*52 =7700.

    So the total number of ways to get any two suited cards is 400+7700 = 8100.

  7. How many ways can the player get two non-suited cards?
  8. There are combin(4,2)=6 ways to pick 2 suits out of 4. Once two suits have been chosen there are 13*5 = 65 in the shoe to choose from. So the total number of ways to get two non-suited cards is 6*652 = 25,350.

     

     

     

     

     

     

  9. Fill in the following table for the side bet West Hollywood Match and determine the expected value.
  10. Hand

    Combinations

    Pays

    Return

    West Hollywood Match*

    100

    50

    5000

    Suited and ranked face card pair

    120

    25

    3000

    Any other 2 suited cards

    8100

    2

    16200

    All other

    25350

    -1

    -25350

    Total

    33670

    -1150

    Expected value = __-3.42%_______________

    Questions 6-12: Consider an 8-deck blackjack side bet based on the playerís first two cards and the dealerís up card. The king of hearts is known as the suicidal king.

  11. How many ways are there to get 3 suicidal kings?
  12. For all questions on this bet note there are 8 suicidal kings in the shoe, 3*8=24 non-suicidal kings, and 48*8 = 384 non-kings.

    Combin(8,3) = 56.

  13. How many ways are there to get any 3 kings, aside from 3 suicidal kings?
  14. 3 non-suicidal: combin(3*8,3) = combin(24,3) = 2024

    2 non-suicidal: combin(24,2)*8 = 276*8 = 2208

    1 non-suicidal: 24*combin(8,2) = 24*28 = 672

    Total = 2024 + 2208 + 672 = 4904

  15. How many ways are there to get 2 suicidal kings, plus one non-king?
  16. Combin(8,2)*(48*8) = 28*384 = 10,752

  17. How many ways are there to get any two kings (besides two suicidal kings), plus one non-king?
  18. 2 non-suicidal: combin(24,2)*384 = 276*384 = 105,984

    1 non-suicidal: 24*8*384 = 73,728

    Total = 105984 + 73728 = 179,712.

  19. How many ways are there to get any one king (plus two non-kings)?
  20. (24+8)*combin(384,2) = 32*73,536 = 2,353,152.

  21. How many ways are there to get zero kings?
  22. Combin(384,3) = 9,363,584.

     

     

     

     

     

  23. Fill in the following table to get the expected value of the side bet Suicidal Kings.
  24. Hand

    Combinations

    Pays

    Return

    Three suicidal kings

    56

    10000

    560000

    Any three kings

    4904

    500

    2452000

    2 suicidal kings

    10752

    100

    1075200

    Any two kings

    179712

    10

    1797120

    Any one king

    2353152

    1

    2353152

    No kings

    9363584

    -1

    -9363584

    Total

    11912160

    -1126112

    Expected value = -1126112/11912160 = -9.45%

  25. You have a sock drawer with 8 white socks and 5 black socks. You pull out 3 at random and without replacement. What is the probability you pull out at least 2 white socks?
  26. 3 whites: combin(8,3) = 56.

    2 white, 1 black: combin(8,2)*5 = 28*5 = 140

    Total: combin(13,3) = 286

    Probability = (56+140)/286 = 68.53%.

    14. Fill in the following table for the game Derby.

    Quinella

    Pays (for one)

    Indifferent probability

    Fair Probability

    Fair Odds (for one)

    1, 2

    8

    0.125000

    0.109589

    9.124990

    1, 3

    34

    0.029412

    0.025786

    38.781208

    1, 4

    22

    0.045455

    0.039851

    25.093723

    1, 5

    2

    0.500000

    0.438357

    2.281248

    2, 3

    70

    0.014286

    0.012524

    79.843664

    2, 4

    46

    0.021739

    0.019059

    52.468694

    2, 5

    4

    0.250000

    0.219178

    4.562495

    3, 4

    200

    0.005000

    0.004384

    228.124755

    3, 5

    17

    0.058824

    0.051571

    19.390604

    4, 5

    11

    0.090909

    0.079701

    12.546862

    Total

    --------

    1.140624

    1.000000

    ---------------

  27. Assuming the house edge is the same on every bet, what is it?

1-(1/1.140624) = 12.33%