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

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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ReplyDeleteThis comment has been removed by the author.

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