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.)
(In reply to
Solution by Penny)
I did write a trial-and-error Visual Basic console application program, not to solve the puzzle, but to test my solution after the fact. The program very strongly supports an answer of 0.078 out of 1.000 as the correct answer. (My posted solution gave the value as 0.07 out of 1.00)
The program:
Imports System
Imports System.Runtime.InteropServices
Imports System.Math
Module Module1
Dim dblRoll1 As Double
Dim dblRoll2 As Double
Dim dblRoll3 As Double
Dim dblVar1 As Double
Dim dblVar2 As Double
Dim dblWinCount As Double
Dim dblLoseCount As Double
Dim dblMoney As Double
Dim dblNumberOfGames As Double
Dim dblIntRsltsCount As Double
Dim dblRunningCount As Double
Dim dblTotalCount As Double
Dim dblWinningOdds As Double
Sub Main()
Randomize()
dblNumberOfGames = 0
dblTotalCount = 0
dblRunningCount = 0
dblIntRsltsCount = 0
While dblNumberOfGames = 0
Console.WriteLine("How many times shall we play ?")
dblNumberOfGames = Int(Console.ReadLine())
End While
While ((dblIntRsltsCount = 0) Or _
(dblIntRsltsCount >= dblNumberOfGames))
Console.WriteLine("After how many game series do you want intermediate" _
& " results ?")
dblIntRsltsCount = Int(Console.ReadLine())
End While
dblWinCount = 0
dblLoseCount = 0
For Index1 As Integer = 1 To dblNumberOfGames
dblMoney = 10
dblTotalCount += 1
dblRunningCount += 1
If dblRunningCount >= dblIntRsltsCount Then
dblRunningCount = 0
Console.WriteLine("Number of game series so far = " & _
Str(dblTotalCount))
dblWinningOdds = dblWinCount / dblTotalCount
Console.WriteLine("We won " & Str(dblWinCount) & " times and lost " & _
Str(dblLoseCount) & " times")
Console.WriteLine("Winning odds were " & Str(dblWinningOdds))
End If
While dblMoney > 0
dblRoll1 = Int(((12 - 1 + 1) * Rnd()) + 1)
dblRoll2 = Int(((12 - 1 + 1) * Rnd()) + 1)
dblRoll3 = Int(((40 - 1 + 1) * Rnd()) + 1)
dblVar1 = (dblRoll1 * dblRoll1) - dblRoll2
dblVar2 = (dblRoll2 * dblRoll2) - dblRoll1
If dblVar1 >= dblRoll3 And _
dblVar2 <= dblRoll3 Then
dblMoney += 10
If dblMoney = 30 Then
dblWinCount += 1
Exit While
End If
Else
dblMoney -= 10
If dblMoney = 0 Then
dblLoseCount += 1
Exit While
End If
End If
End While
Next
Console.WriteLine(" ")
dblWinningOdds = dblWinCount / dblNumberOfGames
Console.WriteLine("Final Results:")
Console.WriteLine("We won " & Str(dblWinCount) & " times and lost " & _
Str(dblLoseCount) & " times")
Console.WriteLine("Winning odds were " & Str(dblWinningOdds))
Console.WriteLine("Press Enter to terminate program")
Console.ReadLine()
End Sub
End Module
Edited on April 18, 2004, 12:05 am
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Posted by Penny
on 2004-04-17 22:28:07 |
re: Solution
