The ∑ simply means sum, therefore to get the correct expected value we need to sum all the
probability multiplied by value for all probable events.
As an example, we can work out the expected value for a single number $1 bet on an
American Roulette table. There 38 possible numbers and we assume each number is equally
likely. Therefore, the probability of winning is 1 in 38 and the probability of losing is 37 in 38.
When we lose, we lose $1. When we win, we win $35.
So our expected value is $-0.0526 or -5.26%. This is also the house edge on an American
roulette table. On a European roulette table, the expected value is $-0.027 or -2.7%. So
expected value tells us that we should always prefer a European Roulette table over an
American Roulette table.
In reality, expected value is not always so easy to calculate. The above method assumes
perfectly random outcomes. This may not always be the case. For example, we can expect a
coin toss game to give heads and tails evenly and work out our expected value accordingly
but we may find by observation that heads comes up 3:1 against tails. In this case, the coin
may be biased to heads because of some flaw in the coin.
Therefore, you should note that the general form of expected value is predictive based on
perfect randomness. Where you suspect that this assumption is not true, then it is better to
use historical data to estimate expected value.
