Coin Flip Sequences 

When flipping a fair coin until either HHT or HTT are first reached, on average it takes longer for HHT to occur than for HTT, even though the sequence probabilities are equal.

In[1]:=

X proc = DiscreteMarkovProcess[ 1, {{1/2, 1/2, 0, 0, 0, 0}, {0, 0, 1/2, 1/2, 0, 0}, {0, 0, 1/2, 0, 1/2, 0}, {0, 1/2, 0, 0, 0, 1/2}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}}];

In[2]:=

X Graph[{"Start", "H", "HH", "HT", "HHT", "HTT"}, proc, ImageSize -> Medium]

Out[2]=

Find the mean number of coin flips to reach HHT.

In[3]:=

X Mean[FirstPassageTimeDistribution[proc, 5]] // N

Out[3]=

Find the mean number of coin flips to reach HTT.

In[4]:=

X Mean[FirstPassageTimeDistribution[proc, 6]] // N

Out[4]=