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What is the expected number of coin flips needed to get two consecutive heads?
This is the same question as problem #12 in the Statistics Chapter of Ace the Data Science Interview!
Let X be the number of coin flips needed to obtain two consecutive heads. We then want to solve for E[X]. Let H denote a flip that results in heads, and T denote a flip that results in tails. Note that E[X] can be written in terms of E[X|H] and E[X|T], i.e., the expected number of flips needed, conditioned on a flip being either heads or tails respectively.
Conditioning on the first flip, we have:
Note that E[X|T] = E[X] since if a tail is flipped, we need to start over in getting two heads in a row.
To solve for E[X|H], we can condition it further on the next outcome: either heads (HH) or tails (HT).
Therefore, we have:
Note that if the result is HH, then E[X|HH] = 0 since the outcome has been achieved, and that E[X|HT] = E[X] since a tail was flipped, we need to start over in attempting to get two heads in a row. Thus:
Plugging this into the original equation yields:
and after solving we get:
Therefore we would expect 6 flips.