HMD436 Test #2 Preparation Name______________________
Consider a three reel single-line slot machine with weighted reels. The following table shows the total weights per reel for each symbol.
|
Symbol |
Reel 1 |
Reel 2 |
Reel 3 |
|
Bar |
4 |
3 |
2 |
|
Bell |
5 |
4 |
3 |
|
Plum |
6 |
5 |
4 |
|
Orange |
7 |
6 |
5 |
|
Blank |
42 |
46 |
50 |
|
Total |
64 |
64 |
64 |
|
Payline |
Combinations |
Probability |
Pays |
Return |
|
3 bars |
24 |
0.000092 |
2500 |
0.228882 |
|
3 bells |
60 |
0.000229 |
1000 |
0.228882 |
|
3 plums |
120 |
0.000458 |
500 |
0.228882 |
|
3 oranges |
210 |
0.000801 |
250 |
0.200272 |
|
Total |
0.001579 |
0.886917 |
Consider a modified keno game in which there are 50 balls in the chamber, of which 12 will be drawn at random. The pick 7 game pays 5 for matching 4, 50 for matching 5, 500 for matching 6, and 10000 for matching 7. Fill in all cells in the following table to determine the player’s expected return.
|
Match |
Combinations |
Probability |
Pays |
Return |
|
4 |
5075297955 |
0.04180653 |
5 |
0.20903264 |
|
5 |
676706394 |
0.00557420 |
50 |
0.27871019 |
|
6 |
42675178 |
0.00035153 |
500 |
0.17576318 |
|
7 |
962598 |
0.00000793 |
10000 |
0.07929166 |
|
Total |
5795642125 |
0.047740188 |
0.7427977 |
The probability of matching x out of 7 balls is combin(7,x)*combin(43,12-x)/combin(50,12).
Consider a game of Derby with the following pay table. As usual for games where the player doesn’t get the original bet back all pays are on a "for one" basis.
|
Quinella |
Pays |
Fair probability |
True probability |
Return |
|
1,2 |
23 |
0.043478261 |
0.03482 |
0.8008614 |
|
1,3 |
47 |
0.021276596 |
||
|
1,4 |
108 |
0.009259259 |
||
|
1,5 |
13 |
0.076923077 |
||
|
2,3 |
7 |
0.142857143 |
||
|
2,4 |
16 |
0.0625 |
||
|
2,5 |
2 |
0.5 |
||
|
3,4 |
32 |
0.03125 |
||
|
3,5 |
4 |
0.25 |
||
|
4,5 |
9 |
0.111111111 |
||
|
Total |
1.248655447 |
What is the player’s expected return on the 1,2 quinella?
0.8008614
Determine the expected value of the Royal Match side bet in a 9-deck game.
|
Hand |
Combinations |
Probability |
Pays |
Return |
|
Royal match |
324 |
0.002964915 |
25 |
0.0741229 |
|
Easy match |
26820 |
0.245429089 |
2.5 |
0.6135727 |
|
Non-suited |
82134 |
0.751605996 |
-1 |
-0.751606 |
|
Total |
109278 |
1 |
-0.0639104 |
Formulas:
Royal match 4*9^2
Easy match 4*((COMBIN(13,2)-1)*9^2+13*COMBIN(9,2))
Non-suited COMBIN(4,2)*(9*13)^2
Determine the expected value of the Pair Square side bet in a 7-deck game.
|
Hand |
combinations |
probability |
pays |
Return |
|
Suited pair |
1092 |
0.016528926 |
15 |
0.2479339 |
|
Non-suited pair |
3822 |
0.05785124 |
10 |
0.5785124 |
|
non-pair |
61152 |
0.925619835 |
-1 |
-0.9256198 |
|
total |
66066 |
1 |
-0.0991736 |
Formulas:
Suited pair 13*4*COMBIN(7,2)
Non-suited pair 13*COMBIN(4,2)*7^2
non-pair COMBIN(13,2)*(4*7)^2
Determine the expected value for the Super Sevens bet in a 5-deck game.
|
Hand |
Permutations |
Probability |
Pays |
Return |
|
1 seven |
1238400 |
0.07128007 |
3 |
0.21384021 |
|
2 unsuited 7's |
72000 |
0.00414419 |
50 |
0.20720951 |
|
2 suited 7's |
19200 |
0.00110512 |
100 |
0.11051174 |
|
3 unsuited 7's |
6600 |
0.00037988 |
500 |
0.18994205 |
|
3 suited 7's |
240 |
0.00001381 |
5000 |
0.06906984 |
|
Non-paying hand |
16037280 |
0.92307692 |
-1 |
-0.92307692 |
|
Total |
17373720 |
1.00000000 |
-0.13250357 |
Formulas:
1 seven (5*4)*(5*48)*(5*52-2)
2 unsuited 7's PERMUT(4,2)*5^2*(48*5)
2 suited 7's 4*PERMUT(5,2)*(48*5)
3 unsuited 7's PERMUT(5*4,3)-4*PERMUT(5,3)
3 suited 7's 4*PERMUT(5,3)
Non-paying hand (48*5)*PERMUT(52*5-1,2)
Three Card Poker Modified Deck Combinations
Let r = number of ranks
Let s = number of suits
Straight flush s*(r-1)
Three of a kind r*combin(s,3)
Straight (r-1)*(s^3-s)
Flush 4*(combin(r,3)-(r-1))
Pair r*(r-1)*combin(s,2)*s
Nothing (combin(r,3)-(r-1))*(s^3-s)
War Modified Deck Probabilities
Let r = number of ranks
Let s = number of suits
Let d = number of decks
Probability of initial tie = r*combin(d*s,2)/combin(d*r*s,2)
Probability of tie given initial tie = ((r-1)*combin(d*s,2)+combin(d*s-2,2))/combin(d*r*s-2,2)
Probabilities in Bingo
Probability of a blackout within x calls = combin(51,x-24)/combin(75,x)
Probability of four corners within x calls = combin(71,x-4)/combin(75,x)
Probability of an X with x calls = combin(67,x-8)/combin(75,x)
Probability of making a specific figure with y marks (not including free square) within x calls = combin(75-y,x-y)/combin(75,x)
|
Number |
Black Out |
4 Corners |
X |
Number |
Black Out |
4 Corners |
X |
|
1 |
0.000000 |
0.000000 |
0.000000 |
38 |
0.000000 |
0.060731 |
0.002899 |
|
2 |
0.000000 |
0.000000 |
0.000000 |
39 |
0.000000 |
0.067671 |
0.003647 |
|
3 |
0.000000 |
0.000000 |
0.000000 |
40 |
0.000000 |
0.075190 |
0.004558 |
|
4 |
0.000000 |
0.000001 |
0.000000 |
41 |
0.000000 |
0.083319 |
0.005663 |
|
5 |
0.000000 |
0.000004 |
0.000000 |
42 |
0.000000 |
0.092089 |
0.006996 |
|
6 |
0.000000 |
0.000012 |
0.000000 |
43 |
0.000000 |
0.101534 |
0.008595 |
|
7 |
0.000000 |
0.000029 |
0.000000 |
44 |
0.000000 |
0.111688 |
0.010505 |
|
8 |
0.000000 |
0.000058 |
0.000000 |
45 |
0.000000 |
0.122584 |
0.012777 |
|
9 |
0.000000 |
0.000104 |
0.000000 |
46 |
0.000000 |
0.134259 |
0.015466 |
|
10 |
0.000000 |
0.000173 |
0.000000 |
47 |
0.000001 |
0.146748 |
0.018639 |
|
11 |
0.000000 |
0.000272 |
0.000000 |
48 |
0.000001 |
0.160089 |
0.022367 |
|
12 |
0.000000 |
0.000407 |
0.000000 |
49 |
0.000002 |
0.174319 |
0.026731 |
|
13 |
0.000000 |
0.000588 |
0.000000 |
50 |
0.000005 |
0.189477 |
0.031822 |
|
14 |
0.000000 |
0.000824 |
0.000000 |
51 |
0.000009 |
0.205603 |
0.037743 |
|
15 |
0.000000 |
0.001123 |
0.000000 |
52 |
0.000017 |
0.222736 |
0.044605 |
|
16 |
0.000000 |
0.001497 |
0.000001 |
53 |
0.000030 |
0.240919 |
0.052535 |
|
17 |
0.000000 |
0.001958 |
0.000001 |
54 |
0.000054 |
0.260193 |
0.061672 |
|
18 |
0.000000 |
0.002518 |
0.000003 |
55 |
0.000097 |
0.280600 |
0.072169 |
|
19 |
0.000000 |
0.003189 |
0.000004 |
56 |
0.000169 |
0.302184 |
0.084197 |
|
20 |
0.000000 |
0.003986 |
0.000007 |
57 |
0.000292 |
0.324991 |
0.097944 |
|
21 |
0.000000 |
0.004924 |
0.000012 |
58 |
0.000498 |
0.349064 |
0.113615 |
|
22 |
0.000000 |
0.006018 |
0.000019 |
59 |
0.000839 |
0.374451 |
0.131436 |
|
23 |
0.000000 |
0.007285 |
0.000029 |
60 |
0.001399 |
0.401197 |
0.151657 |
|
24 |
0.000000 |
0.008742 |
0.000044 |
61 |
0.002306 |
0.429351 |
0.174549 |
|
25 |
0.000000 |
0.010408 |
0.000064 |
62 |
0.003762 |
0.458962 |
0.200408 |
|
26 |
0.000000 |
0.012300 |
0.000093 |
63 |
0.006077 |
0.490078 |
0.229559 |
|
27 |
0.000000 |
0.014439 |
0.000132 |
64 |
0.009723 |
0.522750 |
0.262353 |
|
28 |
0.000000 |
0.016846 |
0.000184 |
65 |
0.015415 |
0.557028 |
0.299174 |
|
29 |
0.000000 |
0.019541 |
0.000254 |
66 |
0.024223 |
0.592966 |
0.340439 |
|
30 |
0.000000 |
0.022547 |
0.000347 |
67 |
0.037743 |
0.630614 |
0.386601 |
|
31 |
0.000000 |
0.025888 |
0.000468 |
68 |
0.058330 |
0.670028 |
0.438147 |
|
32 |
0.000000 |
0.029586 |
0.000623 |
69 |
0.089439 |
0.711260 |
0.495609 |
|
33 |
0.000000 |
0.033667 |
0.000823 |
70 |
0.136103 |
0.754367 |
0.559559 |
|
34 |
0.000000 |
0.038155 |
0.001076 |
71 |
0.205603 |
0.799404 |
0.630614 |
|
35 |
0.000000 |
0.043079 |
0.001395 |
72 |
0.308404 |
0.846427 |
0.709441 |
|
36 |
0.000000 |
0.048464 |
0.001794 |
73 |
0.459459 |
0.895496 |
0.796757 |
|
37 |
0.000000 |
0.054338 |
0.002288 |
74 |
0.680000 |
0.946667 |
0.893333 |
|
75 |
1.000000 |
1.000000 |
1.000000 |