GAM470 Name_____Answers________
March 15, 2005 Homework
Questions 14: Consider a 5deck blackjack game based on the player’s first two cards.
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*5^{2} = 100
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
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 (5212)*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*(781)*5^{2} =7700.
So the total number of ways to get any two suited cards is 400+7700 = 8100.
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 nonsuited cards is 6*65^{2} = 25,350.
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 612: Consider an 8deck 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.
For all questions on this bet note there are 8 suicidal kings in the shoe, 3*8=24 nonsuicidal kings, and 48*8 = 384 nonkings.
Combin(8,3) = 56.
3 nonsuicidal: combin(3*8,3) = combin(24,3) = 2024
2 nonsuicidal: combin(24,2)*8 = 276*8 = 2208
1 nonsuicidal: 24*combin(8,2) = 24*28 = 672
Total = 2024 + 2208 + 672 = 4904
Combin(8,2)*(48*8) = 28*384 = 10,752
2 nonsuicidal: combin(24,2)*384 = 276*384 = 105,984
1 nonsuicidal: 24*8*384 = 73,728
Total = 105984 + 73728 = 179,712.
(24+8)*combin(384,2) = 32*73,536 = 2,353,152.
Combin(384,3) = 9,363,584.
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%
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 
 
1(1/1.140624) = 12.33%