March 8, 2005 Homework

For problems 1 to 8 consider a side bet on a single-deck blackjack game.

1. How many combinations are there for a suited ace/jack blackjack?
2. 4 (one for each suit).

3. How many combinations are there for a non-suited ace/jack blackjack?
4. Number of aces: 4

Number of jacks: 4

Total combinations of all ace/jack blackjack: 4*4=16

Total combinations of non-suited ace/jack blackjacks: 16-4=12

5. How many combinations are there for any other suited blackjack?
6. Number of suits: 4

Number of ranks for 10-pt card: 4

Total suited blackjacks: 4*4=16

Total higher paying hands (suited ace/jack): 4

Total other suited blackjacks: 16-4=12

7. How many combinations are there for all other blackjacks (non-suited and non-ace/jack)?

Total aces: 4

Total 10-pt cards: 16

Total blackjacks: 4*16=64

Total higher paying blackjacks: 4+12+12=28

Total "other" blackjacks: 64-28=36

5. How many combinations are there for non-blackjacks?

Both cards 2 to 9 = combin(32,2) = 496

Two aces = combin(4,2)=6

Two 10-pt cards = combin(16,2) = 120

One ace and one 2-9 = 4*32 = 128

One 10-pt card and one 2-9 = 16*32 = 512

Total = 496+6+120+128+512 = 1262

1. How many combinations are there of any two cards in a single deck?
2. Total combinations = combin(52,2) = 1326

3. Based on the answers to 1-5 determine the total return for the game for all combinations.
4.  Hand Combinations Pays Return Suited ace/jack BJ 4 50 200 Other ace/jack BJ 12 25 300 Other suited BJ 12 25 300 Any other BJ 36 10 360 Non-BJ 1262 -1 -1262 Total 1326 -102
5. Divide the total return by the total combinations to get the expected value of this side bet.
6. -102/1326 = -7.69%.

7. You have a drawer with drawer with 10 white socks and 6 black socks. You reach in and draw 5 at random without replacement. What is the probability you draw a full house (three of one color and two of the other)?
8. 3 whites and 2 blacks: combin(10,3)*combin(6,2) = 120*15 = 1800

2 whites and 3 blacks: combin(6,3)*combin(10,2) = 20*45 = 900

Total combinations: combin(16,5) = 4368

Probability = (1800+900)/4368 = 61.81%.

9. A keno game has 50 numbers. The casino will draw 10 of them. You pick 5 numbers. What is the probability you catch exactly 2 of them?
10. Ways to draw 2 winning balls out of 10 = combin(10,2) = 45

Ways to draw 3 losing balls out of 40 = combin(40,3) = 9880

Total 2 out of 5 combinations = 45*9880 = 444,600.

Total all combinations = combin(50,5) = 2,118,760.

Probability = 444,600/2,118,760 = 20.98%.

Problems 11-99: Consider a side bet called "Fantastic Fives." The game uses five decks of 52-cards each. The player is paid according to his own initial two cards and the dealer’s up card.

11. How many combinations are there for three suited fives?
12. Number of suits: 4

Number of ways to choose 3 fives of a particular suit out of 5 in the shoe = combin(5,3) = 10.

Total suited five combinations: 4*10 = 40

13. How many combinations are there for three non-suited fives?
14. Total fives in the shoe = 4*5 = 20

Total ways to draw 3 fives out of 20 = combin(20,3) = 1140

Suited fives = 40 (see problem #11)

Non-suited fives = 1140-40=1100

15. How many combinations are there for two fives, which are suited?
16. Number of suits: 4

Number of ways to choose 2 fives out of the 5 in the shoe of a particular suit = combin(5,2) = 10

Number of ways to draw the non-5 singleton = 48*5 = 240

Total combinations = 4*10*240 = 9600.

17. How many combinations are there for two fives, which are not suited?
18. Number of ways to draw two fives out of 20 in the shoe = combin(20,2) = 190.

Number of ways to draw the non-5 singleton = 48*5 = 240

Total ways to draw any two fives (including suited) = 190*240 = 45,600

Total ways to draw two non-suited fives = 45,600 – 9600 = 36,000

19. How many combinations are there for one five?
20. Ways to get one five = 20

Ways to get two non-fives out of the 48*5=240 in the deck = combin(240,2) = 28,680.

Total combinations = 20*28680 = 573,600

Fill in the following table to find the expected value of the game, which is the total return divided by the total combinations.

 Hand Combinations Pays Return Three sutied fives 40 2000 80000 Three fives 1100 200 220000 Two suited fives 9600 50 480000 Two unsuited fives 36000 20 720000 One five 573600 1 573600 No fives 2275280 -1 -2275280 Total 2895620 -201680
21. What is the expected value?

-201680/2895620 = -6.97%.