Expected value and variance in plain English
Why one session can swing far above or below the average: a plain-English look at expected value and variance, using roulette as the example.
Expected value (EV) and variance are two ideas that explain most of what you see in casino results. EV tells you the average result of a bet if it could be repeated many times. Variance tells you how far real results swing around that average.
Expected value
To work out EV, multiply each possible result by its chance and add them up. Take a straight-up bet of 1 unit on a single-zero roulette wheel. It wins 35 units with a chance of 1 in 37 and loses 1 unit with a chance of 36 in 37. That gives (35 x 1/37) - (1 x 36/37) = -1/37, which is -2.70% of the stake.
On a 10-unit bet, the EV is about -0.27 units. Every bet on a single-zero wheel has the same EV of -2.70%, because the payouts are set as if the wheel had 36 numbers while it has 37 pockets. On a double-zero wheel the figure is -5.26%.
A negative EV is the house edge seen from the player's side. It describes the average, not the next spin.
Variance: the swings around the average
Two bets can have the same EV and feel very different. A red or black bet wins 18 times in 37 (48.65%) and pays 1 to 1, so results are small and frequent. A straight-up bet wins 1 time in 37 and pays 35 to 1, so results are rare and large.
Imagine 37 straight-up bets of 1 unit. On average you expect one win, and an average loss of 1 unit overall. In practice:
- With no win, which happens about 36% of the time, you are down 37 units.
- With exactly one win, you are down 1 unit.
- With two wins, you are up 35 units.
The average is the same, yet the three results are very far apart.
Now take 100 even-money bets of 1 unit. The expected loss is 2.70 units, but a typical swing is around 10 units either way. That is why roughly one run of 100 such bets in three can finish ahead even though every bet is negative on average.
Short term and long run
In the short run, variance dominates, and luck decides who is ahead. In the long run, the average result per bet moves towards the EV. This does not mean results "even out". The total loss tends to grow with the number of bets, because the edge applies to every unit staked.
Being ahead after a session does not show that you found a better method. It shows that variance went your way, and it can go the other way next time. No betting system can turn a negative EV into a positive one.
18+ only. This article is for information, not advice to gamble. Responsible gambling.
