Two students play a game based on the total roll of two standard dice. Student A says that a 12 will be rolled first. Student B says that two consecutive 7s will be rolled first. The students keep rolling until one of them wins. What is the probability that A will win?
Solution:
Let p be the probability that student A wins. We consider the possible outcomes of the first two rolls. (Recall that each roll consists of the throw of two dice.) Consider the following mutually exclusive cases, which encompass all possibilities.
Probability p is the weighted mean of all of the above possibilities.
Hence p = 1/36 + 1/216 + (29/216)p + (29/36)p.
Therefore p = 7/13.
Solution:
Let p be the probability that student A wins. We consider the possible outcomes of the first two rolls. (Recall that each roll consists of the throw of two dice.) Consider the following mutually exclusive cases, which encompass all possibilities.
- If the first roll is a 12 (probability 1/36), A wins immediately.
- If the first roll is a 7 and the second roll is a 12 (probability 1/6 · 1/36 = 1/216), A wins immediately.
- If the first and second rolls are both 7 (probability 1/6 · 1/6 = 1/36), A cannot win. (That is, B wins immediately.)
- If the first roll is a 7 and the second roll is neither a 7 nor a 12 (probability 1/6 · 29/36 = 29/216), A wins with probability p.
- If the first roll is neither a 7 nor a 12 (probability 29/36), A wins with probability p.
Probability p is the weighted mean of all of the above possibilities.
Hence p = 1/36 + 1/216 + (29/216)p + (29/36)p.
Therefore p = 7/13.
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ReplyDeleteUse This code in your microsoft office 2010 and you will get p = 1/2
ReplyDeleteSub DiceGameTest()
Randomize
Dim NewValue1 As Single
Dim NewValue2 As Single
Dim LastValue1 As Single
Dim LastValue2 As Single
Dim StudentAwin As Single
Dim StudentBwin As Single
Dim Winner As Single
For i = 1 To 5000000
LastValue1 = 0
LastValue2 = 0
Winner = 0
While Winner = 0
LastValue1 = NewValue1
LastValue2 = NewValue2
NewValue1 = Int((6 * Rnd) + 1)
NewValue2 = Int((6 * Rnd) + 1)
If NewValue1 = 6 And NewValue2 = 6 Then Winner = 1
If NewValue1 + NewValue2 = 7 And LastValue1 + LastValue2 = 7 Then Winner = 2
Wend
If Winner = 1 Then StudentAwin = StudentAwin + 1
If Winner = 2 Then StudentBwin = StudentBwin + 1
Winner = 0
Next i
total = StudentBwin + StudentAwin
Selection.TypeText Text:=str(StudentAwin / total)
End Sub