Question

You have two number cubes. One number cube has faces (1,2,2,3,3,4) and the other has faces (1,3,4,5,6,8). You areconsidering a game in which you win 100 tokens If the sum is greater than or equal to 8 but lose 80 tokens If the sum isless than 8. Should you play this game?

Answer 1

To determine whether or not to play this game, the following steps are necessary:

Step 1: Draw up a table that shows the sum of the outcomes when the two cubes are tossed together, as follows:

The table above shows that there are a total of 36 outcomes when the two cubes are tossed together, and shows the sum of the values on the faces of the cubes for each outcome.

Step 2: Use the values in the table to find the probabiity of obtaining a sum greater than or equal to 8, and the probability of obtaining a sum less than 8, as below:

$\begin{gathered} P(sum\text{ is }\ge8)=\frac{\text{total number of sum values greater than or equal to 8}}{\text{total number of outcomes}} \\ \text{From the table:} \\ \text{total number of sum values greater than or equal to 8 = 15} \\ \text{total number of outcomes = 36} \\ \text{Thus:} \\ P(sum\text{ is }\ge8)=\frac{15}{36} \end{gathered}$

Also:

$\begin{gathered} P(sum\text{ is <}8)=\frac{\text{total number of sum values less than 8}}{\text{total number of outcomes}} \\ \text{From the table:} \\ \text{total number of sum values less than 8 = 21} \\ \text{total number of outcomes = 36} \\ \text{Thus:} \\ P(sum\text{ is <}8)=\frac{\text{2}1}{\text{3}6} \end{gathered}$

Step 3: Compute the expectation, using the probabilities and the tokens to be won or lost, as follows:

$\begin{gathered} \text{Total expectation = (+100 tokens)}\times P(sum\text{ is }\ge8)\text{ + (-80 tokens)}\times P(sum\text{ is <}8) \\ \text{Thus:} \\ \text{Total expectation = (+100 tokens)}\times\frac{15}{38}\text{ + (-80 tokens)}\times\frac{21}{36} \\ \text{Total expectation =}\frac{1500}{38}\text{ + }(\frac{-1680}{36}) \\ \text{Total expectation =}\frac{1500+(-1680)}{38}=\frac{1500-1680}{36}=-\frac{180}{36}=-5 \\ \Rightarrow\text{Total expectation =}-5\text{ tokens} \end{gathered}$

Now, since the total expectation is -5 tokens, it means that 5 tokens will be lost at the end of this game. Now, would you play a game where you get to lose? Certainly not.

The answer is that you should not play this game