Expectation is the return that can be anticipated from any given wager or proposition after enough bets have been made to normalize the result.

All experienced poker players are intimately acquainted with the concepts of both positive and negative fluctuation. Results from any given proposition may vary greatly from a correctly calculated expectation in the short term, but will completely normalize themselves after a sufficient sampling.

We know that the odds against drawing an Ace from the top of a well and fairly shuffled deck are 12:1 against.

Suppose that you are offered a proposition where you will lose $100 every time the first card off is not an Ace but you will be paid 15:1 every time the card is an Ace.

While you will lose 12 of 13 tries under the terms of this proposition, you still have a nice positive expectation and the only 2 things that can stop you from winning a large long-term profit are

  1. You stop taking the bet
  2. You don't have the bankroll to withstand the inevitable fluctuations.

On average, over the course of 13 such bets you will lose 12 for a loss of $1,200 and win 1 worth $1,500, for a net profit of $300 over 13 tries.

12 losses @ $100 ea = $1,200 loss 1 Win @ $1,500 = $1,500 profit Net Profit = $300

With a profit of $300 over 13 tries, you can anticipate an average profit of $300 / 13 = $23.08 per trial. The value or expectation of each draw is +$23.08.

The sum of all expectations in any proposition must be zero.

You, as the player, have a positive expectation of +$23.08 per draw and the person who laid the bet has a negative expectation of -$23.08.

Fluctuation is short-term.

After 20 draws it would not be unusual for you, as the taker of such a wager, to be down $2,000 and your opponent, who made the proposition, up $2,000 — a fluctuation. While after a small number of tries you might be down, it is almost a 100% certainty that after 100,000 tries you will be up over $2,000,000 and your opponent will be down over $2,000,000 — an expectation.

Expectation is long-term.

The same applies to every bet or raise you make or take playing poker. As long as all of your bets, calls and raises carry greater money odds than risk odds, you will be a long-term winner.

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