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# Expected value

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In a series of coin flips, a run is a series of consecutive coin flips that are all the same. For example, in the sequence
the red letters form a run.

If a fair coin is flipped two times, what is the expected length of the longest run?

Mar 15, 2023

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There are four possible outcomes when flipping a fair coin two times: HH, HT, TH, and TT, where H represents heads and T represents tails. Let's consider each of these cases separately:

- HH: In this case, the longest run is 2, since both flips are heads.
- HT: In this case, the longest run is 1, since there is a run of heads of length 1 and a run of tails of length 1.
- TH: Same as HT.
- TT: In this case, the longest run is 2, since both flips are tails.

So the possible lengths of the longest run are 1 and 2. To find the expected length, we can calculate the probability of each outcome and weight it by the length of the corresponding run:

- P(longest run is 1) = P(HT or TH) = 2/4 = 1/2
- P(longest run is 2) = P(HH or TT) = 2/4 = 1/2

So the expected length of the longest run is:

E(longest run) = 1*P(longest run is 1) + 2*P(longest run is 2)
= 1*(1/2) + 2*(1/2)
= 1.5

Therefore, the expected length of the longest run when flipping a fair coin two times is 1.5.

Mar 16, 2023