Expectation Values

Once you can calculate the probability of an event (using a plain relative requency, a probability tree, Bayes Theorem or a distribution functions), you may need to account for the value of the possible outcomes. This is called calculating an expectation value.

Example

You roll a six-sided die.

• if you roll a 6, you get 10 points

• if you roll a 1, you lose 5 points

• for any other number you lose 1 point

How many points do you win or lose on average?

Generic Definition

The expectation value E of the event X is the sum of the rewards of each possible outcome \(v_i\) times their possibilities \(p(X_i)\):

\[E(X) = \sum _i v_i \cdot p(X_i)\]

Applied to the example above, we consider the outcomes 6, 1 and other:

\[E(roll) = 10 \cdot p(six) - 5 \cdot p(one) - 1 \cdot p(two-to-five)\]

\[E(roll) = 10 \cdot \frac{1}{6} - 5 \cdot \frac{1}{6} - 1 \cdot \frac{4}{6}\]

\[E(roll) = \frac{10}{6} - \frac{5}{6} - \frac{4}{6} = \frac{1}{6}\]

Conclusion: On average, you win one-sixth of a point per roll.

Adding Normal Distributions

A frequent application of expectation values is when values are sampled from more than one normal distribution. In this case, you need to add the mean and standard deviation to get the resulting distribution functions.

Example

A Chinstrap and a Gentoo penguin jump on a balance. What is the probability that their combined weight is more than 8000 g?

• the weight of Chinstrap penuins is normally distributed with a mean of 3700 g and a standard deviation of 380 g.

• the weight of Gentoo penuins is normally distributed with a mean of 5000 g and a standard deviation of 500 g.

The combined weight is also normally distributed, with a mean of 8700 g and a standard deviation of 880 g.

Thus, the probability that the pair is heavier than 8000 g can be calculated in a spreadsheet with:

=1 - NORM.DIST(8000, 8700, 880, TRUE)

which should result in about 78%.

Subtracting Normal Distributions

A similar rule applies to the difference of two distributions. In this case, you would subtract the means of both distributions but still add the standard deviations.

Example

What is the probability that the Gentoo penguin is 1000 g heavier than the Chinstrap penguin?

Using the same distributions as above, the difference Gentoo - Chinstrap follows a normal distribution with a mean of 1300 g (5000 g - 3700 g) and a standard deviation of 880 g.

The probability that the difference is 1000 g or higher can be calculated in a spreadsheet with:

=1 - NORM.DIST(1000, 1300, 880, TRUE)

which should result in about 63%.