MATHCOUNTS State Target 2025 (Problem 8)
Xing has an unfair coin with probability of getting heads. What is the expected number of flips he will need to get a head followed by two tails in three consecutive flips? Express your answer as a common fraction.
Define 3 variables and 3 equations to solve the problem.
Define , , and .
= the expected number of flips needed to obtain HTT.
= the expected number of additional flips needed to obtain HTT, given that the most recent flip was H.
= the expected number of additional flips needed to obtain HTT, given that the two most recent flips were HT.
= the expected number of flips needed to obtain HTT.
= the expected number of additional flips needed to obtain HTT, given that the most recent flip was H.
= the expected number of additional flips needed to obtain HTT, given that the two most recent flips were HT.
Calculate .
Multiply the equations in Hints 4 and 5 by 3 and simplify.
Find .