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 \(\frac{7}{2}\).