Question 1117249
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The problem is worded poorly, making different interpretations possible.  However, since the question is<br>
"What is the probability that a player WILL GET OUT OF JAIL by throwing a double on the next 3 turns?"<br>
I would agree with you that it is only asking the probability that the player will get out of jail by throwing a double on ONE of the next three turns.<br>
However, neither calculation you show is correct for the respective interpretations.<br>
For your friend's interpretation, that the player gets doubles on all three of the next turns, the probability is the calculation you show for YOUR interpretation: (1/6)*(1/6)*(1/6) = 1/216.<br>
For your interpretation, the probability has to interpreted as
(1) get a double on turn 1, OR
(2) not get a double on turn 1 AND get a double on turn 2, OR
(3) not get a double on turn 1 AND not get a double on turn 2 AND get a double on turn 3<br>
Replacing the ORs with addition and the ANDs with multiplication, the calculation for that probability is<br>
{{{(1/6)+(5/6)(1/6)+(5/6)(5/6)(1/6) = (1/6)+(5/36)+(25/216) = (36+30+25)/216 = 91/216}}}<br>
Notice the probability can also be calculated as 1 minus the probability that he does NOT get a double on any of the next three turns:<br>
{{{1 - (5/6)(5/6)(5/6) = 1 - 125/216 = 91/216}}}