March 15, 2005 Homework

Questions 1-4: Consider a 5-deck blackjack game based on the player’s 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
• Suited jack and king

Expected value = __-3.42%_______________

Questions 6-12: Consider an 8-deck blackjack side bet based on the player’s first two cards and the dealer’s 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%