Using Expected Values to Make Decisions

Statistics and probability are some of the most useful fields within math. Why? Because they can help us make all kinds of real-world decisions. One concept in probability that can be especially useful in decision-making is expected value. Let's review the meaning of expected value and find out why it can help us come to decisions.

What is expected value?

You might recall that expected value is the weighted average of all possible values in our experiment. Weighted averages are results multiplied by the probability of these results occurring.

We can use the expected value of a random variable to make important decisions - such as whether to enter a competition, whether to play a game, or whether to make an investment.

An example of how expected value can help us make decisions

Once we know how to calculate weighted values and expected values, it's easy to find out whether certain decisions are worth making.

For example, consider a special game of roulette. Unlike a normal game of roulette, this roulette has numerous colors and just six different spaces. Of these spaces, there are two yellow spaces, one red space, two green spaces, and one blue space. Take a look:

Before you enter, the person running the game tells you that if the ball lands on a red square, you win two tokens. If the ball lands on blue, you win 5 tokens. Any other result will lose you 1 token.

Now -- can we use our knowledge of probability to find out whether this game is worth playing?

Let's start with the basics: the roulette wheel has six segments and there's an equal chance of the ball landing on each of these segments.

The sample space is {red sector, blue sector, all other sectors}

Now let's write a probability distribution:

The chance of landing on red (2-token win) or blue (5-token win) is $\frac{1}{6}$ . The chance of landing on all other segments (1 token loss) is $\frac{4}{6}$ .

Let's plug those values into our formula for expected value:

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

$=\frac{2}{6}+\frac{5}{6}-\frac{4}{6}$

$=3/6or0.5$

So what does this mean? If our expected value is 0.5 it means that we stand to win an average of 0.5 tokens per each spin (or one token per two spins). Those are good odds, and if we keep spinning, we would keep winning tokens.

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