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What is Probability?

In common usage, the word "probability" means the possibility of some event.

It is a well-known fact now that the frequency of outcomes is a stable value in a long trial period. Many experimentalists tossed an unbiased coin and counted the frequency of "head" occurrences. Every time when the number of trials was great enough the frequency of "head" occurrences was about 0.5. For example, the next table shows the results obtained by French naturalist Buffon (1707-1788) in the XVIII century and results obtained at beginning of XX century by English statistician Pirson (1857-1936).

The effect described above and recurrent observations of other mass social and natural phenomena are the ground for the next conclusion. The frequency of event outcomes is nearly equal to some constant value in a long trial period. For example, the frequency of boys' birth is 0.518 nd the frequency of girls' birth is 0.482.

This constant value is called probability.

Significance of Expected Value

Expected value (EV) is a central principle in the theory of probability. It is used for average estimation of some random value. Expected value is similar to a center of gravity assuming that the values of probability are the masses of solid point.

Let us assume that the game has n different outcomes, probability of each outcome is p_i. The expected value of some variable x that takes values x_i can be calculated with the next formula:

E(x) = x₁p₁ + x₂p₂ + ... + x_np_n ,

In case the probabilities of outcomes are equal ( p₁=p₂=...=p_n=1/n ) the expected value equals to arithmetic mean:

E(x) = x₁/n+ x₂/n + ... + x_n/n = (x₁ + x₂ + ... + x_n) / n

Why is the expected value the most important principle in the probability theory? It helps to predict the estimation of some random variable for a long period of trials. The mean of any random variable in a long term gets close to its expected value. This fact is strictly proved in the course of probability theory.

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.

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.