Dicey Probability
Satvik and Agnishom are playing a game with
standard dice. Both the dice are rolled together and the total is counted.
Satvik says that a total of will be rolled first.
Agnishom, whereas, says that two Consecutive totals of will be rolled first.
They keep rolling the dice till one of them wins!
The probability that Satvik wins the game can be expressed as where and are co-prime positive integers.
Find the value of .
The answer is 20.
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Let the probability that Satvik wins the game be P.
There are 5 possibilities.
1) If the first roll is a 2 (probability 1/36), Satvik wins immediately.
2) If the first roll is a 7, and the 2nd roll is a 2, (1/6 *1/36 = 1/216) Satvik wins immediately
3) If the first two rolls are both 7, (1/6 *1/6= 1/36) Satvik loses.
4) If the first roll is a 7 and the 2nd roll is neither a 7 nor a 2, (1/6 * 29/36 = 29/216), Satvik is back to square one and can win with probability P.
5) if the first roll is neither a 7 nor a 2, (29/36), Satvik is again back to square one and can win with probability P.
Probability P is the weighted mean of all the above possibilities,
P = 1/36 + 1/216 + (29/216) P + (29/ 36) P
Solving P = 7/13
Answer is 7 + 13 = 20