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# plz help

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In a sequence 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.
$$TT \textcolor{red}{HHH} TTHHHTH,$$
If a fair coin is flipped four times, what is the expected length of the longest run?

Feb 2, 2023

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Because we are dealing with 4 coins, the easiest way (imo) is to list out all 16 cases and count the average

The 16 cases are:

HHHH   HTHT   TTTT   THTT

HHHT   HTTH   TTTH   THHT

HHTH   HTTT   TTHT   THTH

HHTT   HTHH   TTHH   THTH

We see that 2 have a run of 4, 4 have a run of 3, 8 have a run of 2, and 2 have a run of 1.

So the expected value is $${{(2 \times 4) + (4 \times 3) + (8 \times 2) + (2 \times 1)} \over 16} = \color{brown}\boxed{2.375}$$

Feb 2, 2023