Toss Coin Until Head Appears Twice In a Row

Take a fair coin and toss it until a head appears twice in a row. There is no finite sample space for this experience, for it theoretically can go on infinitely. That said, the first few elements are: {H,H}, {T,H,H}, {T,T,H,H}, {H,T,H,H}. To determine the probability of tossing exactly 4 times, we can take the elements in the sample space that corresponds to only 4 tosses. In that case, they are {T,T,H,H}, {H,T,H,H}. The probability of tossing either combination is 1/16, and two such combination gives a total probability of 1/8.

To see the result simulated, the following codes on MATLAB can help to illustrate that with 1 million trials. Documentations are added to clarify the algorithm.

probability = 0.5;      %prob of tossing head
numTrials = 1000000;    %number of simulation trials
trialResults = zeros(1,numTrials);

for trial = 1:numTrials
    done = false;
    lastHead = false;
    toss = 0;
    while ~done         %this loop controls each trial
        roll = rand;
        toss = toss + 1;
        if roll < probability && lastHead
            trialResults(trial) = toss;
            done = true;    %each trial over when last trial was H, and this trial is also
        elseif roll < probability && ~lastHead
            lastHead = true;    %last trial was not H, but this trial is
        else
            lastHead = false;   %did not toss a H
        end
    end
end

sum(trialResults == 4) / numTrials

The results vary very little from 0.1250.