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Streak of heads somewhere

ProbabilityHard

A fair coin is tossed $6$ times, with independent tosses. What is the probability that at least $3$ heads in a row appear somewhere in the sequence?

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Hint

Count the complement: sequences with no streak of $3$ heads, by looking at how a sequence ends.

Solution

Let $b (n)$ be the number of sequences of length $n$ with no $3$ heads in a row.
Such a sequence ends in tails, in tails then one heads, or in tails then $2$ heads, so $b (n) = b (n - 1) + b (n - 2) + b (n - 3)$ .
The start values are $b (1) = 2$ , $b (2) = 4$ and $b (3) = 7$ .
Then $b (4) = 13$ , $b (5) = 24$ and $b (6) = 44$ .
The sequences with a streak number $64 - 44 = 20$ , so the probability is $\frac{20}{64}$ , which is $\frac{5}{16}$ .

Answer: $\frac{5}{16}$