# Expected Values

When betting on an uncertain outcome, it helps to think in terms of the average profit that results if the same bet in the same situation is made repeatedly. It is useful to define the expected value of a bet as the average profit that results after many repetitions. Expected values can be positive, negative or zero, depending on whether the bet wins money over the long-run, loses money, or breaks even. The expected value of a bet depends on the odds against success and the payoff if the bet succeeds.

For example, consider having four cards to a flush, with one card to come, and it costs $1 to contest a $5 pot. The odds against success are 4 to 1, which means on average, four out every five times, this bet will lose. It costs $5 to make this bet five times and its one success will return $6—the $5 already in the pot, plus the $1 put in to contest it. The ratio of 6 to 5 is 1.2. That means the expected value of the bet is $1.2 -$1 or $0.20. The bet expects to return a profit $0.20 per dollar on average every time it is placed.

Consider the same situation with the cards, but with a $3 pot. It now costs $1 to contest a $3 pot. Again, it will cost $5 to make this bet 5 times, but the one success will return $4. The ratio of 4 to 5 is 0.8, so the expected value is $0.8 - $1 or -$0.20. On average, this bet will lose $0.20 per dollar each time it is placed.

The expected values is a useful concept but keep in mine the following:

- Bets with positive expected values can lose just as often as bets with negative expected values. For the examples just given, the winning frequencies are the same. A bet that is "good" in a mathematical sense might lose most of the time.
- Conversely, bets with negative expected values can win just as frequently as bets with positive expected values.
- Bets that are "bad" in a mathematical sense do win a certain fraction of the time.
- For a bet to have a positive expected value, the payoff must be greater than the odds against success.
- The expected value is an average of many repetitions; it is not the outcome of a single event. For each hand, usually you either win or you lose. Fractional outcomes, such as split pots, are rare.
- To win over the long-run at poker, you must consistently place bets with positive expected values and avoid ones with negative expected values.
- The expression "on average" does not mean that if the odds against success are 4 to 1, every five bets placed will always include one success. If poker were that predictable, no one would play the game.