Please help!
H/T, $50\%$-$50\%$ independent events.
I pick Heads.
What is the probability that after $29$ attempts of a fair coin, there's no consecutive Heads (...HH...) at all?
I would appreciate seeing how it's worked out...
Thanks a bunch.
2026-04-02 09:10:26.1775121026
Probability of no two consecutive heads (or tails) after n trials?
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First you need to know how many different sequences there are which don't have two consecutive heads. (Then you can just multiply this by the probability of a specific sequence - $(1/2)^{29}$ - to get the probability.)
Hint
Write $a_n$ for the number of sequences of length $n$ without HH. If the sequence starts with a tail, then the remaining $n-1$ terms just have to avoid two consecutive heads, so there are $a_{n-1}$ sequences which start with a tail. How many sequences start with a head?