I set up three states: start, one head seen, done.
Define states based on the current streak of consecutive heads (0, 1, or 2) and set up equations for the expected number of flips from each state. Solve the system of linear equations to find the expected flips from the start state.
Pro tip: After solving, verify the result using an alternative method like martingale or simulation to demonstrate robustness and catch errors.
Identify the relevant states: S0 (no consecutive heads, i.e., last flip was T or no flips), S1 (last flip was H but not two in a row), and S2 (two consecutive heads, absorbing state).
Let E0, E1, and E2 be the expected additional flips from each state. Write equations: E0 = 1 + 0.5 E0 + 0.5 E1, E1 = 1 + 0.5 E0 + 0.5 E2, and E2 = 0.
Substitute E2 = 0 into the equations and solve for E0 and E1. From E1 = 1 + 0.5 E0, and E0 = 1 + 0.5 E0 + 0.5 E1, derive E0 = 6.
Check the solution by plugging back or using an alternative method (e.g., simulation or martingale). Interpret the result: on average, it takes 6 flips to see two consecutive heads.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Same state setup, just replace 0.5 with p and 0.5 with (1-p).
Set up a system of equations using states based on the current run of consecutive heads (0, 1, or 2). Solve for the expected number of flips from state 0, then verify with a simpler case (e.g., p=0.5) to ensure correctness.
Pro tip: Always check your general formula against known special cases (like p=0.5 giving 6 flips) to catch algebra mistakes and demonstrate rigor.
Define states based on the current streak of consecutive heads: state 0 (no recent heads), state 1 (one head), state 2 (two heads, absorbing).
Let E0, E1, E2 be the expected additional flips from each state. Write equations: E2=0, E1=1 + p*E2 + (1-p)*E0, E0=1 + p*E1 + (1-p)*E0.
Solve for E0 in terms of p. From E0 equation: E0 = 1/p + E1. Substitute E1 = 1 + (1-p)E0, then solve to get E0 = (1+p)/p^2.
Check p=0.5 gives 6, p=1 gives 2, and p→0 gives infinity, confirming the formula's plausibility.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Model the process as a Markov chain with states: 0 (no progress), 1 (last flip was H), 2 (HH achieved). Set up equations for the expected value and second moment of the hitting time T, then compute Var(T) = E[T^2] - (E[T])^2. Alternatively, use the martingale or pattern-matching approach to directly derive the variance.
Pro tip: After deriving the variance, mention that the distribution of T is not geometric and has a longer tail; this shows deeper understanding. Also, be prepared to generalize to other patterns or biased coins.
Define states: S0 (no progress), S1 (last flip H), S2 (HH achieved). Specify transition probabilities: from S0, H→S1, T→S0; from S1, H→S2, T→S0; S2 is absorbing.
Let E_i be expected flips to reach S2 from state i. Write E_0 = 1 + 0.5 E_0 + 0.5 E_1, E_1 = 1 + 0.5*0 + 0.5 E_0, E_2=0. Solve to get E_0 = 6.
Let M_i = E[T^2 | start in state i]. Use the relation M_i = 1 + sum_j P_ij (2 E_j + M_j) for i≠2, with M_2=0. Write equations for M_0 and M_1.
Solve the linear system for M_0 and M_1. Then Var(T) = M_0 - (E_0)^2. Compute the final numerical value.
Check that variance is positive and reasonable. Optionally, compare with simulation or known results. Discuss implications for the distribution of T.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.