[Math] How to go about calculating the expected profit/winning per play of this game

combinatoricsprobability

I designed this game for my Data class but I'm having alot of trouble calculating the expected profit per play of this game. We have taken up examples involving lottery tickets in which you have to calculate the profit and winning per ticket bought but I can't seem to apply the same concepts to my game due to the various stages and rules.

This is how the games goes

You pay $20 to start and there are three rounds

Round 1: There is a 1/2 chance you win and move on, and a 1/2 chance you lose

Round 2: There is a 2/6 chance you move on to the next round, and 1/6 chance you win $10 AND move on. (3/6 chance you lose and get nothing)

Round 3: There is 1/12 chance you win $100, and if you lose you get nothing

Here is my attempt at it

Chart

After multiplying every correlating x and y value i get E(X) = -18.5, meaning the profit the ones running the game make is $18.5

Help is greatly appreciated!!!

Best Answer

You pay $\$20$ and start playing. Let your profit be $x$ which is initially $-20$.

  • With a probability of $\frac12$, you lose: $x = -20$.

  • Otherwise, with a probability of $\frac12$, you proceed with unchanged $x$.

    • With a probability of $\frac12$, you lose: $x = -20$.

    • With a probability of $\frac16$, you proceed with a new $x = -10$.

      • With a probability of $\frac{11}{12}$, you lose: $x = -10$.
      • Otherwise, with a probability of $\frac{1}{12}$, you have $x = 90$.
    • Otherwise, with a probability of $\frac13$, you proceed with unchanged $x$.
      • With a probability of $\frac{11}{12}$, you lose: $x = -20$.
      • Otherwise, with a probability of $\frac{1}{12}$, you have $x = 80$.

Summing along this tree of probabilities, $$\begin{align*}E(x) &= \frac12 (-20) + \frac12 \bigg(\frac12 (-20) + \frac16\bigg(\frac{11}{12}(-10) + \frac{1}{12}(90) \bigg) + \frac13\bigg(\frac{11}{12}(-20) + \frac{1}{12}(80) \bigg)\bigg) \\ &= -\frac{205}{12} = -17\,{}^1/_{12}\end{align*}$$

Thus you will expect to lose $\$17\,{}^1/_{12}$ in this game, which is the profit of the people running the game. Your calculation probably went wrong somewhere, the best way to avoid this is to make a tree like I did (perhaps more graphically a `tree').

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