Problem:
Misha rolls a standard, fair six-sided die until she rolls -- in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is , where and are relatively prime positive integers. Find .
Solution:
Call a sequence of dice rolls an even sequence if -- first occurs as the last three rolls of a sequence of an even number of rolls, and an odd sequence if it first occurs as the last three rolls of a sequence of an odd number of rolls. Let be the probability that Misha rolls an odd sequence. Let be the conditional probability that Misha rolls an odd sequence given that her first roll is a , and let be the conditional probability that Misha rolls an odd sequence given that her first two rolls are -. Misha can roll an odd sequence either by rolling a on her first roll with probability and then completing an odd sequence with probability , or by rolling something other than a on her first roll with probability followed by rolling an even sequence with probability . Similarly, given that Misha's first roll is a , she can roll a on her second roll and then, ignoring the first roll, complete an even sequence with probability ; she can roll a on her second roll and then complete an odd sequence with probability ; or she can roll another number followed by an odd sequence with probability . Finally, given that Misha's first two rolls are -, she can roll a on her third roll and then, ignoring the first two rolls, complete an odd sequence with probability ; roll a ; or roll another number followed by an even sequence with probability . Thus
This system can be solved by substitution to get , and . The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions