Sneaky Joe has just invited you as a VIP to his new casino. You know this is probably an attempt steal your money, for he always find ways to swindle people. However, you go anyway.

When you get there, he says, "Come over here and join me in a game of craps." You become slightly suspicious, but agree to come anyway. When you go over, he says, "OK, here's how we play craps in this casino, 'cause it's different here than other casinos. You have 3 dice, 2 of them are 12-sided dice and another is a 40-sided die. I will roll the 2 12-sided dice. Then you roll the 40-sided one. If your number is between (y^2-x) and (x^2-y) inclusively, being that x=the number I got from the first roll and y=the number I got on the second, you will win $10. Otherwise, you will lose $10."

"Ok," you think, "I'm pretty sure that the odds are against me, especially if it's a game that Joe made himself. But I need $30, and I only have $10." So, what's the probability of you winning $30 (as in $30 in the black, without any debt, which included the original $10 paid) from this game?

(NOTE: It can be done WITHOUT trial and error, and it is my request, though you do not have do it, that you solve this without trial and error.)

Despite the request of the author, (sorry) I used Excel to permutate all 144 possible die rolls, calculate (x^2)-y and (y^2)-x, and then went down the list to tabulate how many numbers 1 through 40 fell between the two function columns.  I then divided their sum by (144*40) which gave me the probability of winning one round at 50.15625%.  Since we need to win two net times, and can not afford to lose one net times, the probabilities are:

WW: 0.251564941
WLW: 0.1253894
the sum of which is 0.376954342

Therefore I calculate the odds of making the requisite $20 profit to bring my total holdings to $30 to be 37.6954342%

x y y^2-x x^2-y #Hits 1-40
1 1 0 0 0
1 2 3 -1 3
1 3 8 -2 8
1 4 15 -3 15
1 5 24 -4 24
1 6 35 -5 35
1 7 48 -6 40
1 8 63 -7 40
1 9 80 -8 40
1 10 99 -9 40
1 11 120 -10 40
1 12 143 -11 40
2 1 -1 3 3
2 2 2 2 1
2 3 7 1 7
2 4 14 0 14
2 5 23 -1 23
2 6 34 -2 34
2 7 47 -3 40
2 8 62 -4 40
2 9 79 -5 40
2 10 98 -6 40
2 11 119 -7 40
2 12 142 -8 40
3 1 -2 8 8
3 2 1 7 7
3 3 6 6 1
3 4 13 5 9
3 5 22 4 19
3 6 33 3 31
3 7 46 2 39
3 8 61 1 40
3 9 78 0 40
3 10 97 -1 40
3 11 118 -2 40
3 12 141 -3 40
4 1 -3 15 15
4 2 0 14 14
4 3 5 13 9
4 4 12 12 1
4 5 21 11 11
4 6 32 10 23
4 7 45 9 32
4 8 60 8 33
4 9 77 7 34
4 10 96 6 35
4 11 117 5 36
4 12 140 4 35
5 1 -4 24 24
5 2 -1 23 23
5 3 4 22 19
5 4 11 21 11
5 5 20 20 1
5 6 31 19 13
5 7 44 18 23
5 8 59 17 24
5 9 76 16 25
5 10 95 15 26
5 11 116 14 27
5 12 139 13 28
6 1 -5 35 35
6 2 -2 34 34
6 3 3 33 31
6 4 10 32 23
6 5 19 31 13
6 6 30 30 1
6 7 43 29 12
6 8 58 28 13
6 9 75 27 14
6 10 94 26 15
6 11 115 25 16
6 12 138 24 17
7 1 -6 48 40
7 2 -3 47 40
7 3 2 46 39
7 4 9 45 32
7 5 18 44 13
7 6 29 43 12
7 7 42 42 0
7 8 57 41 0
7 9 74 40 1
7 10 93 39 2
7 11 114 38 3
7 12 137 37 4
8 1 -7 63 40
8 2 -4 62 40
8 3 1 61 40
8 4 8 60 33
8 5 17 59 24
8 6 28 58 13
8 7 41 57 0
8 8 56 56 0
8 9 73 55 0
8 10 92 54 0
8 11 113 53 0
8 12 136 52 0
9 1 -8 80 40
9 2 -5 79 40
9 3 0 78 40
9 4 7 77 40
9 5 16 76 25
9 6 27 75 14
9 7 40 74 1
9 8 55 73 0
9 9 72 72 0
9 10 91 71 0
9 11 112 70 0
9 12 135 69 0
10 1 -9 99 40
10 2 -6 98 40
10 3 -1 97 40
10 4 6 96 35
10 5 15 95 26
10 6 26 94 15
10 7 39 93 2
10 8 54 92 0
10 9 71 91 0
10 10 90 90 0
10 11 111 89 0
10 12 134 88 0
11 1 -10 120 40
11 2 -7 119 40
11 3 -2 118 40
11 4 5 117 36
11 5 14 116 27
11 6 25 115 16
11 7 38 114 3
11 8 53 113 0
11 9 70 112 0
11 10 89 111 0
11 11 110 110 0
11 12 133 109 0
12 1 -11 143 40
12 2 -8 142 40
12 3 -3 141 40
12 4 4 140 37
12 5 13 139 28
12 6 24 138 17
12 7 37 137 4
12 8 52 136 0
12 9 69 135 0
12 10 88 134 0
12 11 109 133 0
12 12 132 132 0

Posted by John on 2004-04-20 16:53:05