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Expected Return and Player Edge

Let us start with the next example. There is a lottery of 1000 tickets; every ticket costs $1. Lottery has the following prizes: one ticket win $500, 5 tickets win $50 and 20 tickets has $10 prize.

Number Probability Prize,$
001 001/1000 500
005 005/1000 050
020
020/1000
010
974
974/1000
000

Therefore, total prize amount is 1*$500 + 5*$50 + 20*$10 = $950, or average $950/1000 = $0.95 for every ticket. This index is called Expected Return (ER).

Now deduct the ticket price from expected return: $0.95-$1=-$0.05. This index is player Edge or profit. It means the average game result. Note that this lottery is not a profitable game; you lose about $0.05 on every ticket.

The example we used above is easy and the calculations are evident. However, there are many other games where we can't apply the same methods. How to find a general principle for expected return or edge calculation?

In general, expected return can be computed as expected value of win, and player edge as expected value of game results. Really, the previous results can be found using this formula.

Expected return is a sum of all products of win to probability
500*1/1000 +50*5/1000 + 10*20/1000 + 0*974/1000 = 0.95

Player edge is a sum of all products of game result to probability
(500-1)*1/1000 + (50-1)*5/1000 + (10-1)*20/1000 + (0-1)*974/1000 = -0.05

For convenience sake the expected return and edge are regularly computed as per cent of wager:
Expected return = 100% * (0.95$/1$) = 95%
Player edge = 100% * (-0.05/1$) = -5%

Therefore, the next formula shows the relation of the expected return and edge in percentage:
Edge = Expected Return - 100%

Usually the edge is used for a basic rating of gamble benefit; but sometimes the expected return is considered as the main index (in video poker).

If you know the player edge you can forecast the gamble productivity for a long period of game. For example, roulette profit is -2.7%. Therefore, if you bet $1 for 1000 times, the game balance will be about -2.7%*$1000 = -$27.

Moreover, the expected return affords to compare gamble efficiency of different games and to choose the best one. It is evident that a game with a higher expected return is more preferable for gambling. Moreover, you can earn money when expected return is positive!

It is so, but casino managers also know how to earn money. There are a few games with player advantage. These games require from player a perfect strategy, experience and high skills. For playing Black Jack, you have to count running cards. If you select video poker, you have to memorize at least 20-30 rules of hand playing with some exceptions.

Only if you master the perfect strategy you can reach the advantage! Otherwise, you have to be a lucky to beat the casino.


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