+0

# Probability

0
79
2

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
+6
0

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

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
+2
+1

There are four possible outcomes when flipping a fair coin two times:

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.

my account access

Feb 20, 2023