Question

Here is a game: you roll a fair die. If the outcome is an odd number you win that amount in dollars and the game is over. If the outcome is an even number, you roll the die again. If on the second roll the outcome is a “6” you win six dollars; otherwise, you lose the amount that the die shows. Let X denote the random variable that states how much you win.

(a) Write down the probability mass function of  X.

(b) Calculate the expected value of  X.

Answer 1

If we get 1, 3, or 5 then the game is over and we win either, $1, $3, or $5 depend on the outcome as 1, 3, 5, respectively.

So the probabilities of the value of X as 1, 3, 5 are 1/6, 1/6, 1/6 respectively.

Next if the outcome is an even number then we roll a die .

Now, if we get 6 then we win $6 with probability (1/6)*(1/2) =

If we get 1, 2, 3, 4, 5, then we loss $1, $2, $3, $4, $5 , respectively with probabilities (1/6)*(1/2) of each.

Let's make table:

The probability distribution of X is as follows:

Let's find expected value of X