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?

TheOneXWZ Feb 2, 2023

#1**+2 **

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 \({{(2 \times 4) + (4 \times 3) + (8 \times 2) + (2 \times 1)} \over 16} = \color{brown}\boxed{2.375}\)

BuilderBoi Feb 2, 2023