GAM470 Name_____Answers_____________
March 8, 2005 Homework
For problems 1 to 8 consider a side bet on a single-deck blackjack game.
4 (one for each suit).
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
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
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
Total combinations = combin(52,2) = 1326
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 |
-102/1326 = -7.69%.
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%.
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.
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
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
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.
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
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 |
-201680/2895620 = -6.97%.