# 4.5 Expectation

## Another Dice And Coin Game

Suppose I flip a fair, two-sided coin. If it comes up heads, then I roll a fair six-sided die until I get an odd number and record the value. Otherwise, I roll until I get an even number and record the value.
Using the law of total expectation, find the expected value of the experiment. Please answer as a decimal with two significant figures.

Exercise 1

Let $$C$$ be an indicator variable for the event that the coin comes up heads and let $$R$$ be the value on the die. Using the Law of Total Expectation, we get $$E[R]=E[R|C]\Pr[C]+E[R|\overline{C}]\Pr[\overline{C}]$$. Firstly, $$\Pr[C]=\Pr[\overline{C}]=\frac{1}{2}$$. Secondly, $$E[R|C] = 1\cdot\frac{1}{3}+3\cdot\frac{1}{3}+5\cdot\frac{1}{3}=3$$ and $$E[R|\overline{C}] = 2\cdot\frac{1}{3}+4\cdot\frac{1}{3}+6\cdot\frac{1}{3}=4$$. Hence, the answer is $$\frac{7}{2}$$.