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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?

 Feb 20, 2023
 #1
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   DinarGuru

To solve this problem, we can start by listing all the possible outcomes of flipping a fair coin two times:

HH (two heads)

HT (one head, one tail)

TH (one tail, one head)

TT (two tails)

We can then calculate the length of the longest run for each outcome:

HH: The longest run is 2.

HT: The longest run is 1.

TH: The longest run is 1.

TT: The longest run is 2.

Therefore, the expected length of the longest run is the average of the longest runs for each outcome:

Expected longest run = (2 + 1 + 1 + 2) / 4 = 1.5

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

 Feb 20, 2023
edited by reneau34  Feb 20, 2023
 #2
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There are four possible outcomes when flipping a fair coin two times:

 

HH (heads on both flips)

HT (heads on the first flip, tails on the second flip)

TH (tails on the first flip, heads on the second flip)

TT (tails on both flips)

Let X be the length of the longest run in a sequence of two coin flips. We can compute the probability of each possible value of X:

X = 1 if the two coin flips are different (HT or TH), which occurs with probability 1/2 + 1/2 = 1.

X = 2 if the two coin flips are the same (HH or TT), which occurs with probability 1/4 + 1/4 = 1/2.

The expected value of X is the sum of the possible values of X multiplied by their respective probabilities:

E(X) = 1*(1/2) + 2*(1/2) = 1.5

Therefore, the expected length of the longest run in a sequence of two coin flips is 1.5.

 

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 Feb 20, 2023

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