PS - Expected Value Lesson

Expected Value

Imagine you are rolling a fair dice, what is the probability of rolling a 1? a 2? a 6? The probability of rolling each number is 1/6.

Roll	1	2	3	4	5	6
Probability	1/6	1/6	1/6	1/6	1/6	1/6

So what value do we expect to roll? Let's calculate the expected value:

$E\left(x\right)=1\left(\frac{1}{6}\right)+2\left(\frac{1}{6}\right)+3\left(\frac{1}{6}\right)+4\left(\frac{1}{6}\right)+5\left(\frac{1}{6}\right)+6\left(\frac{1}{6}\right)=3.5$

What does 3.5 mean? It means that if we were to record each roll many times, the average would be 3.5.

How did we calculate it? We took each outcome and multiplied by the probability of that outcome and added those together. Officially, the formula looks like this:

$E\left(x\right)=\begin{matrix} n \\ \sum \\ i=1 \end{matrix} =x_{1} p_{1}$

What if we had a "weighted" die? Let's say I gave you a six-sided weighted die and shared the probability of getting each number (see the table). Calculate the expected value.

Value	Probability
1	0.2
2	0.2
3	0.11
4	0.38
5	0.08
6	0.03

E(X)= 1(0.2) + 2(0.2) + 3(0.11) + 4(0.38) + 5(0.08) + 6(0.03)=3.03

OK, so let's consider a fair die again - and let's say that we're playing a game that cost $2 to play. Here are the possible ways to win:

• Roll an even number - win $1
• Roll a 5 - win $5
• Roll a 1 or 3 - lose

Let's analyze the possible outcomes. It is important that when we consider the outcomes, we remember that we already paid $2 to play the game.

Roll	Roll an even number	Roll a 5	Roll a 1 or 3
Monetary Outcome	-1	3	-2
Probability	$\frac{3}{6}=\frac{1}{2}=0.5$	$\frac{1}{6}=0.1667$	$\frac{2}{6}=\frac{1}{3}=0.333$

The expected value is: $-1\left(\frac{1}{2}\right)+3\left(\frac{1}{6}\right)+-2\left(\frac{1}{3}\right)={$}{-0.67}$

So, on average you will lose money and probably shouldn't play this game too much!

Expected Value Practice

IMAGES CREATED BY GAVS