Question 1191771
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She gets a Yahtzee in any of three ways; calculate the probability in each case.<br>
(1) both dice show 4 on the second roll<br>
P(4) on each die is 1/6, so P(4 on both dice) is (1/6)(1/6)=1/36<br>
(2) one die shows 4 on the second roll and the remaining one die shows 4 on the third roll<br>
P(4) on each die is 1/6; P(not 4) on each die is 5/6.
Either the first die can be a 4 and the other not, or the other way around.
P(4 then not 4) = (1/6)(5/6) = 5/36
P(not 4 then 4) = (5/6)(1/6) = 5/36
P(1 of 2 dice showing 4) = 10/36 = 5/18
P(4 on third roll) = 1/6
P(Yahtzee with one 4 on second roll and one on third roll) = (5/18)(1/6) = 5/108<br>
(3) neither die shows 4 on the second roll and both show 4 on the third roll.<br>
P(neither die shows 4 on second roll) = (5/6)(5/6) = 25/36
P(both dice show 4 on third roll) = (1/6)(1/6) = 1/36
P(Yahtzee with two 4's on third roll - (25/36)(1/36) = 25/1296<br>
P(Yahtzee with three 4's on first roll) = (1/36)+(5/108)+(25/1296) = (36/1296)+(60/1296)+(25/1296) = 121/1296<br>