Solution
Waiting for Double Heads
Show the problem again
You flip a fair coin repeatedly. What is the expected number of flips until you see two heads in a row?
Worked solution
The answer is 6. Let E be the expected flips from scratch and E_H the expectation given the last flip was heads. Then E = 1 + (1/2)E_H + (1/2)E and E_H = 1 + (1/2)(0) + (1/2)E. Solving: E_H = 1 + E/2 and E = 2 + E_H, so E = 2 + 1 + E/2, giving E = 6.
Source: Standard renewal or Markov-chain waiting-time exercise from probability texts. Statement written for AxiomIQ.