dice game

wtrow

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Jan 24, 2011
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You have a single die. If you roll a 2 or 4 you win the game, if you roll a 6 you lose. Otherwise, keep rolling until you get a 3 or 6. If you get a 3 you win and a 6 you lose. What is the probability you win?


I know on the first roll the probability is 2/6 you win, 1/6 you lose, and 3/6 you continue. Onwards its 1/6 win, 1/6 loss, and 4/6 roll again. We've never done a continuous problem like this one so I'm completely lost.
 
Hello, wtrow!

You have a single die.
If you roll a 2 or 4 you win the game; if you roll a 6 you lose.
Otherwise, keep rolling until you get a 3 or 6.
If you get a 3, you win. if you roll a 6, you lose.
What is the probability you win?

I know on the first roll the probability is 2/6 you win, 1/6 you lose, and 3/6 you continue.
Onwards its 1/6 win, 1/6 loss, and 4/6 roll again. . Correct!

Let's baby-step through this . . .

Win on 1st roll
. . P(win 1st roll)=26=13\displaystyle P(\text{win 1st roll}) \:=\:\tfrac{2}{6} \:=\:\tfrac{1}{3}

Win on 2nd roll
P(cont. 1st roll)=36=12\displaystyle P(\text{cont. 1st roll}) \,=\,\tfrac{3}{6} \,=\,\tfrac{1}{2}
P(win 2nd roll)=16\displaystyle P(\text{win 2nd roll}) \,=\,\tfrac{1}{6}
. . P(Win 2nd roll)=1216\displaystyle P(\text{Win 2nd roll}) \:=\:\tfrac{1}{2}\cdot\tfrac{1}{6}

Win on 3rd roll
P(cont. 1st roll)=12\displaystyle P(\text{cont. 1st roll}) \,=\,\tfrac{1}{2}
P(cont. 2nd roll)=23\displaystyle P(\text{cont. 2nd roll}) \,=\,\tfrac{2}{3}
P(win 3rd roll)=16\displaystyle P(\text{win 3rd roll}) \:=\:\tfrac{1}{6}
. . P(Win 3rd roll)=122316\displaystyle P(\text{Win 3rd roll}) \:=\:\tfrac{1}{2}\cdot\tfrac{2}{3}\cdot\tfrac{1}{6}

Win on 4th roll
P(cont. 1st roll)=12\displaystyle P(\text{cont. 1st roll}) \,=\,\tfrac{1}{2}
P(cont. 2nd roll)=23\displaystyle P(\text{cont. 2nd roll}) \,=\,\tfrac{2}{3}
P(cont. 3rd roll)=23\displaystyle P(\text{cont. 3rd roll}) \,=\,\tfrac{2}{3}
P(win 4th roll)=16\displaystyle P(\text{win 4th roll}) \,=\,\tfrac{1}{6}
. . P(Win 4th roll)=12(23)216\displaystyle P(\text{Win 4th roll}) \:=\:\tfrac{1}{2}\cdot\left(\tfrac{2}{3}\right)^2\cdot\tfrac{1}{6}

Do you see the pattern?


We have:   P(Win)  =  13+12 ⁣ ⁣16+12 ⁣ ⁣23 ⁣ ⁣16+12 ⁣(23)2 ⁣16+12 ⁣(23)3 ⁣16+\displaystyle \text{We have: }\;P(\text{Win}) \;=\;\frac{1}{3} + \frac{1}{2}\!\cdot\!\frac{1}{6} + \frac{1}{2}\!\cdot\!\frac{2}{3}\!\cdot\!\frac{1}{6} + \frac{1}{2}\!\left(\frac{2}{3}\right)^2\!\frac{1}{6} +\frac{1}{2}\!\left(\frac{2}{3}\right)^3\!\frac{1}{6} + \cdots

. . . . . . . . . . . . . =  13+12 ⁣ ⁣16[1+23+(23)2+(23)3+]geometric series\displaystyle =\;\frac{1}{3} + \frac{1}{2}\!\cdot\!\frac{1}{6}\underbrace{\bigg[1 + \frac{2}{3} + \left(\frac{2}{3}\right)^2 + \left(\frac{2}{3}\right)^3 + \cdots \bigg]}_{\text{geometric series}}

The sum of the geometric series is:   1123=113=3\displaystyle \text{The sum of the geometric series is: }\;\frac{1}{1-\frac{2}{3}} \:=\:\frac{1}{\frac{1}{3}} \:=\:3


Therefore:   P(Win)  =  13+112(3)  =  13+14  =  712\displaystyle \text{Therefore: }\;P(\text{Win}) \;=\;\frac{1}{3} + \frac{1}{12}(3) \;=\;\frac{1}{3} + \frac{1}{4} \;=\;\frac{7}{12}

. . . . exactly as Denis promised!

 
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