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Poker Variance and $ EV Adjusted

Variance is a statistical measure that shows how much the short-term values may vary from the expected mean value (or long-term average in other words). There is no need to be afraid of this language of statistics as the variance will be explained clearly based on the examples. Additionally we will address a way of how we can measure whether we are lucky or unlucky by the $ EV Adjusted.

Variance

Poker is about skill, not about chance. Most poker players agree on this, especially the professionals. This can be denied only by unsuccessful players who blame chance and misfortune for their own mistakes. The well-known saying holds true in poker: "If you always complain about bad luck, then you are not a good player." But let us move ahead and have a look at variance (VAR) and how it is interconnected.

Even though poker is a game of skill, chance plays a partial part of that. The fact that somebody is a better player than some other does not mean that the better player will always win all hands. The worse player may be lucky and beat the better one. However it is not possible to be lucky all the time and thus the better play must prevail as the time goes. That is what it is all about: short run and long run, or in other words, small or big numbers of hands played. In the short term the chance may play a bigger part, but it is suppressed in the long term by player's skill or good play or good decisions if you like.

If you play a sufficiently big number of hands, then you determine (or it gets crystalized by that) your average long-term profit or return on investment (ROI). To make this less easy, it is arguable, what a sufficiently big number of hands is. Even professional players may end up earning nothing after 100,000 hands. However it is for sure that the longer the row of hands played is, the more the regularities occur that establish player's long-term profitability and decrease the variance.

The variance in poker represents the short-term deviations from the player's expected long-term average rate of profit. Again it means that I can lose even though I play well (that is I make a good decision based on (positive) expected value) and on the contrary I can win against the odds. That is what needs to be counted on. Some variance will always be present, but it will be decreased or suppressed in the long term.

→ See an example of poker variance calculation and its practical impact (9 player SNG).

A poker player should not panic when the variance occurs, but hold on and be patient as the good play must manifest itself positively as the time goes. On the other hand the poker player should be critical and sincere with himself and do not mix up his mistakes (or bad play) with the variance. Is there actually a way to measure fortune and misfortune in poker? The answer will be brought by the next chapter—Dollar EV Adjusted.

$ EV Adjusted

Let us first have a look at the key element, that is the Expected Value or EV in short. It does nothing else than assess how much you can win and with what probability and how much you can lose and again with what probability. EV relates to the long-term period and it is a sort of an estimate of mean ("average") expected earnings.

The good poker decisions are those with the positive EV (the maximum EV is desired if possible), often shortened as +EV. The opposite are the bad decisions with the negative expected value or -EV. The higher the EV, the better it is for a player. It will be best explained by the following example.

Example of +EV, -EV

Let us have the following poker situation.

You have in your hand and these cards are on the board. Thus you have a flush draw. The probability to complete the flush is about 0.2 or 20% if you like. The pot is $100 and your opponent bet $25 on turn. The others have folded. Is it worth calling?

Let us count the EV. In words: you can win the pot $125 ($100 + $25) with the 20% probability and you can lose your call $25 with the 80% probability. Then:

EV = 0.2 × $125 + 0.8 × (-$25) = $25 - $20 = +$5.

Conclusion: as the expected value is positive (+EV), it is a good decision to call. Every time this situation occurs, you are likely to win $5 in average in the long-term period. Again, it does not mean that you would win $5 every time, but the wins and losses in this situation would head towards this expected result!

Let us modify our example a bit, be it that the opponent bets $50 instead of $25. Would it still be worth calling in this changed situation? Let us count the EV again. While the probabilities (or odds) of winning and losing are the same, the dollars that can be won $150 ($100 + $50) and lost ($50) changed:

EV = 0.2 × $150 + 0.8 × (-$50) = $30 - $40 = -$10.

Conclusion: If our opponent bet $50, then it would not be worth calling as the EV is negative (-EV). In every situation like this we would be likely to lose $10 in average in the long run. Thus the best decisions would be to fold.

An appendix to the calculation of the probability. How did we actually arrive at the 20% winning possibility? There are 13 hearts in the card deck and we know about 4 of them, hence there are 9 hearts remaining (the hidden cards of the opponent are not to be considered as he can hold any cards, hearts or non-hearts). The whole deck counts 52 cards and 8 of them have been already dealt (2 to me, 2 to the opponent, 4 on the board), hence there are 44 cards remaining in the deck. The probability of completing the flush draw is then 9/44 = 0.2045 = 20.45%.

And Now What Is $ EV Adjusted

$ EV Adjusted is a way to measure luck or bad luck in poker. It is one of the methods, not a perfect one as anything in the life, but it works very well. Dollar EV Adjusted aligns the already realized profits and losses with their probabilities. It is an analogue to the above example of +EV/-EV, but the assessment is ex-post, after hand(s) are played, thus it is not a decision making of what to do any more. Again it is best explained by an example.

Example of $ EV Adjusted
Let us have a specific poker situation—a showdown with no possibility to influence the course of play—in which you have the odds of winning 80% (or 0.8) and the pot is $100.

Your $ EV Adjusted is 0.8 × $100 = $80. This can be considered as a fair value that you are entitled for with the odds given. In other words it is "fair" to gain $80 in this situation. However if you win, you will get the whole pot $100, hence +$20.

The odds of losing in this same situation is 20% (100% – 80%). The risk of losing with regard to the odds can be quantified as 0.2 × $100 = $20. If you were to lose the hand under these circumstances, it would be fair to lose $20 only. But in reality you lose $100, which is -$80 as compared to the "fair" value.

Conclusion
If the $ EV Adjusted is positive, then you have been lucky, if the $ EV Adjusted is negative, then the Fortune has shown you back. You can add up the $ EV Adjusted values for a specific course of hands and find out, whether, according to the expected average return, you have been lucky or unlucky.

From all of the above on this page the following conclusion can be drawn: A player who wants to be successful in poker must focus on maximization of the expected value (EV) and on minimization of the variance (VAR), i.e. the deviations from the expected profitability.