Solution: 2015 Fall Final - 20

Author: Michiel Smid

Question

You repeatedly, and independently, flip three fair coins, until there are exactly two heads among the three flips. Define the random variable $X$ to be the total number of coin flips. For example, if the coin flips result in $$ (TTT), (THT), (HHH), (HTH), $$ then $X = 12$.
What is the expected value $\mathbb{E}(X)$ of $X$?

(a)

(b)

3/8

(c)

(d)

8/3

Solution

Let $X_i$ be 1 if the 3 coin flips result in exactly 2 heads and 0 otherwise.

The probability of getting exactly 2 heads is $ Pr(X_i = 1) = \frac{3}{8} $

Since $X_i$ represents 3 coin flips, we need to divide by 3 to get the expected value of 1 coin flip.

$ E(X_i) = \frac{1}{3} \cdot \frac{3}{8} $

$ E(X_i) = \frac{1}{8} $

Now, we use geometric distribution to find the expected value of $X$.

$ E(X) = \frac{1}{ \frac{1}{8}} = 8 $