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EXPECTED VALUE DEFINED



EV: Expected Value Defined

2006 Randy Saylor

EV is a term thrown around by poker experts (and some less-than-experts), but what is it and how is it applied to poker? Expected value simply is a computation of the amount of money you can expect to win or lose given the average results of each possible outcome.

Flip A Coin

If you and a friend bet on the outcome on coin flips, you can easily apply expected value calculations to better understand the concept. Assuming a fair coin and a sufficiently random coin toss, the likelihood of the coin showing either heads or tails is equal. The probability of each result is 0.50 or 50 per cent. The proper odds should be 1-1, or even money.

You and your friend agree to flip a coin 100 times. She pays you $1 for each “heads,” and you pay her $1 for each “tails.” If you flip the coin 100 times, normal variance might suggest that one of you will come out a few dollars ahead, but if you repeat this trial over a thousand, or even a million, instances, the probability is very high that both of you will end up very close to even.

Your expected value on each coin flip is zero. On average, you will lose $1 and win $1 in equal proportions, resulting in a break-even proposition. If you perform the trial, you are gambling that your side will predominate in the short term. If you agree to perform the trial one billion times, the gamble is reduced, since your long-term results will be much closer to zero. We make no consideration here for the economic value of your time.

That’s easy to understand, but suppose your friend is exceptionally gullible or just doesn’t like money very much. She offers you 2-1 odds that the coin will show heads, but agrees to accept even money for each tails. She puts out $2, and you put out $1. You flip the coin. If it is heads, you take her two dollars. If it is tails, she takes your dollar.

For every two flips on average, you will gain $2 and lose $1, for a net win of one dollar. Dividing the one dollar by the two trials yields the average win per trial, which is 50 cents. The 50 cents is your expected value for each flip of the coin.

You are said to have a positive expected value (+EV) of 50 cents on every flip, and your friend has a negative expected value (-EV) of fifty cents as well. Obviously you should be willing to take a +EV situation an infinite number of times. The larger the EV, the more willing you should be to do so, since a sufficiently large number of trials will reduce the effect of short-term bad luck.

EV of A Loss

Ironically, if you lose a single trial, your EV will always be higher than the amount lost. This takes into account the fact that there is some probability that you will not lose the trial, but instead win money. If the coin in the above example comes up tails, you lose $1, despite the fact that you had a +EV of fifty cents. The fact that you lost money does not change the historical fact that you had +50 cents expected value.

See A Movie

The movie theater example illustrates the EV of a loss. Consider: if you spend $18 on tickets, you’ll likely end up $18 down. But there is a remote chance that you will find a $20 bill on the floor, in which case you had a positive result of $2. If you don’t then take the $20 to the overpriced concessions, you have come out ahead. What if you met a stranger who ended up offering you a job at double your current wage? Although it is impossible to calculate the change in expected value that such a random occurrence would produce, it is always true that the expected loss was smaller than the actual loss.

Assume for a moment that it is possible to calculate the EV of going to the movies, and we calculate it at -$17.90. If you have a normal movie trip, and “lose” $18, your expected loss ($17.90) was smaller than your actual loss ($18).
 

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