Solution: 2014 Winter Final - 22

Author: Michiel Smid

Question

I flip a fair coin, independently, 10 times, resulting in a sequence of heads ($H$) and tails ($T$). For each $HT$ in this sequence, you win \$3. Define the random variable $X$ to be the amount of dollars that you win. For example, if the sequence is $$ HHTTHTTHTT, $$ then $X = 9$. What is the expected value of $X$?

(a)

30/4

(b)

29/4

(c)

27/4

(d)

28/4

Solution

Let $X_i$ be 1 if the current coin flip and the next coin flip are $HT$, and 0 otherwise.

$ \sum_{i=1}^{9} 3 X_i $

The probability of getting HT is $ Pr(X_i = 1) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} $

Since we are determining the current coin flip and the next coin flip, we do not measure the final coin flip and a non-existant next coin flip.

Because we flip the coin 10 times, we can get 9 HTs.

$ 27 \cdot \frac{1}{4} $

$ = \frac{27}{4} $