Random variables. Discrete random variable. Mathematical expectation

Expectation is the average amount of winnings you can expect when playing bookmaker bets. This is probably the most important criterion to follow when choosing a bookmaker. The article is devoted to the calculation of this criterion.

Mathematical expectation is the amount of winnings that a player can expect when placing bets on the same odds.

Example: in a coin game you bet $10 on heads and expect to make a profit of $11, with each successful result your expectation is 0.5. This means that if you bet $10 on tails all the time, then in the long run you will win $0.50 from each bet.

Calculation of mathematical expectation

The formula for calculating this criterion is very simple. First, we multiply the probability of winning by the size of the winnings in each bet. Then we subtract the probability of loss from the result obtained and multiply it by the amount of loss for each bet.

• probability of winning * size of bet payout - probability of loss * amount of bet loss

• First, find the odds for each outcome: draw, win, loss.
• To calculate the potential winnings, multiply the bet amount by the decimal odds for each outcome and then subtract the bet amount.
• To determine the probability of an event, divide 1 by the decimal odds of the outcome we are calculating. The probability of losing is equal to the sum of the probabilities of winning for the other two outcomes.
• Next, we substitute the values ​​we obtained into the formula above.

Example: match between Manchester and Wigan. Odds are 1.263 and 13.5 respectively and a draw with odds of 6.5. If you bet $10 on Vigan, you can win $125. The probability of such an event will be 0.074 or 7.4%. (1 needs to be divided by 13.5).

The odds of another outcome are the sum of the probability of Manchester winning and a draw, in other words 0.792+0.154=0.946. The amount of possible losses is equal to our bet - $10. Our final formula will look like this:

• (0,074 *125$) – (0,946 * 10$) = -0,20$

Judging by the formula, our mathematical expectation here will be negative. We will lose on average $0.20 on every bet.

Why is calculating expected value useful when betting?

A negative mathematical expectation does not mean that we will always lose. All odds in bookmakers are one-sided, which means if we manage to beat the bookmaker, we can win.

How to beat the bookmaker? One way is to calculate with your own hands all the probabilities of a particular sporting event. By calculating the probabilities of events yourself, you can find the bookmaker’s mistake and skillfully take advantage of it. This will significantly increase your chances of winning.

Example: judging by the odds, the chances of Vigan winning are 7.4%. But if, by your calculations, Vigan wins 10% of the time, then our expectation from a bet on him will increase to $3.26.

Calculating the mathematical expectation gives us additional information about bookmakers. Most bookmakers have an expected value of -$1 for every $10 bet, and if you find a positive expected value, you can beat that bookmaker.

Terms from mathematical statistics and probability theory are widely used when playing in a bookmaker's office.

And in general mathematics is an essential science for a prudent gambler.

On the pages of our resource, we try to consider a thoughtful approach to risky investments. This means that the player’s actions on bets must be determined calculations and tactics.

One of the key concepts in game theory is expected value. It helps to evaluate the success of additions in the long term. And it is in this way that those who consider betting their business are going to earn money.

Formula for determining checkmate expectations when playing at a bookmaker's office

When playing bets, the term mathematical expectation can be formulated as follows:

Let us denote familiar concepts with symbols:

• S – bet amount
• W – probability of victory
• L – probability of defeat
• S*k - S – winning size
• M - mat. Expectation

Let's write down the formula for calculating the mathematical expectation:

М=(S*k - S) *W – S*L

The mathematical expectation allows you to evaluate the benefit in the long term, that is, with a sufficiently large number of events.

If mat. expectation is greater than zero, then the player should remain in the black if less than zero, then at a loss.

This statement is valid for a large number of events. That is, positive swearing. waiting does not mean winning on one specific bet. It means a positive balance in a larger number of events. That is, every time you place bets you need to focus on the value of the expectation mate, it must be positive.

Probability of winning and losing

These are two parameters that are subjective when playing bets.

The ability to correctly determine the probability of success of a given team distinguishes a good player from a bad one.

Mathematical calculation example. expectations on bets

Consider the example of a hockey match Sibir - Dynamo, betting on the outcome of the match. Siberia is among the leaders of its division, plays well at home, Dynamo is doing well, and their roster is stronger by name, but the situation with the coach is unclear.

You evaluate many other factors and decide that the probability of victory for Siberia is 60% (0.6) - Dynamo is 40% (0.4).

Bookmakers provide odds for possible outcomes:

• 1.75 - victory of Siberia
• 2.05 - Dynamo win

Let us assume that in this case you decided to bet on Siberia. The bet size is 100 rubles. Let's calculate the math. expectation:

M=(100*1.75-100)*0.6- 100*0.4
M=45-40=5

Mathematical expectation is positive, which means in the long run, you can make a profit. If you can correctly assess the probability of the outcome of matches.

Let's consider the option bets on Dynamo. The bet size, probabilities, odds are the same. Calculate the math. Expectation for this case:

M=(100*2.05-100)*0.4- 100*0.6
M=42-60=-8

Mathematical expectation is negative, which means that with a large number of events, making such bets the player will remain in the red.

It turns out that a good player must take into account several key mathematical factors:

• the probability of the desired outcome,
• bookmaker odds,
• bet size.

And the player’s task is to correctly predict the probability of the occurrence of a particular event. The numerical values ​​of these probabilities are, of course, subjective; it is their determination that is the most difficult and most important factor for success when playing bets.

Here you can always talk about unpredictability, the will of heaven and a high degree of randomness. In such a situation, you often want to rely on at least some knowledge and have at least a little predictability regarding the possibility of winning. It is most often customary to invite higher mathematics to help, namely the concept of mathematical expectation.

The easiest way to talk about it is with examples. This term comes from probability theory and it will be clear to anyone who has studied higher mathematics. Thanks to mathematical calculations, it is possible to obtain results that are not entirely obvious from a mathematical point of view. It turns out that partly apparent randomness can be regulated by mathematical laws. The mathematical expectation is a calculation of the average value of a random variable, i.e. in a vacuum abstract situation it is possible to calculate the probability using it. In particular, the probability of winning. However, when it comes to the lottery, everything is not so simple.

It is important to understand: despite the fact that with the help of mathematical calculations one can easily predict events in which there is no human choice, the anthropogenic factor somewhat changes this picture. And you should approach it with caution. It is worth planning and making calculations based only on the theory of probability with extreme caution. It is possible to calculate the probability of getting the required numbers only in an abstract situation, divorced from reality.

One American mathematics professor, who is an expert in probability theory, sneered at the idea that probability theory has no memory. This means that the prospect of winning the lottery is approximately the same for all players. It is this idea that, as a rule, encourages all participants in such entertainment. There are always chances to win, and using mathematical expectation you can calculate how (not) great they are. And although this is not a guarantee and despite the limitations of the method of use, you can try to work with it. The main thing to consider is that with any number of training sessions, it will be impossible to predict how the game will end in each specific case.

There is a fairly common example of how, in addition to mathematical expectation, a human factor intervenes in a lottery. It is enough to imagine a situation in which a person is offered to play - and this can only be done once - in the lottery. There are two options to choose from.

• In the first, the player is guaranteed to be paid a thousand euros.
• In the second, the player has a fifty percent chance of winning two thousand euros, another forty percent that they will pay out a thousand euros, and there is a ten percent chance that the player will be left with nothing.

In the first version of the lottery the prize is one thousand euros, in the second it is more - one thousand four hundred. Given the obvious benefits of the second option, hardly anyone will doubt that a significant number of participants in the experiment will choose the first option - less profitable, but guaranteed to be reliable. That is why theoretical reasoning will not always have a direct correlation with the practical conclusion and the decision made.

Expectation is also used in other types of random number games. We are talking about all varieties with a strategic component, where, despite the presence of random distribution, the result is still largely influenced by the player’s tactics. The mathematical expectation in such games allows you to competently “manage randomness”, but does not become the main tool.

If we summarize the above information, we can conclude: mathematical probability is one of the factors of probable victory or loss in the lottery, but it alone cannot become a decisive trump card for the player, since other factors are even more important, partly randomness, partly marketing strategy one or another lottery company.

Mathematical expectation (ME) is the sum of the product of the probabilities of making a profit from a transaction, multiplied by the actual result of each trade:

Where n is the number of trades.

Unprofitable trades are substituted into the formula with a negative sign and subtracted when summing, so the expectation takes on both positive and negative values.

The probabilities of a positive outcome (or risk) for each trade are replaced by its actual value, adding the ratio of the arithmetic average of profit and loss. In this case, the formula looks like this:

Where the actual probability is equal to the real percentage of profitable trades from the total number of completed trades.

The average profit is calculated as the sum of profitable transactions divided by their number. The average loss (average loss) is also calculated by summing up the negative values ​​and averaging the results of trades.

The relationship between a flat and a trend changes unpredictably, so it is impossible to accurately calculate the probability when directional movements that have grown to a maximum will bring an amount of loss that cannot be “worked off” with small takes.

Rule for collecting statistical data to calculate the mathematical expectation of profit

Calculations of mathematical expectation are considered reliable if:

the data includes a historical period from 2000 to 10,000 candles or "working time frame" bars; tests equally contain areas of rising, falling trends and flats; volatility does not significantly deviate from historical values ​​(there are no crisis phenomena or panic sales).

Tactical techniques for increasing the value of mathematical expectation

The mathematical expectation strongly depends on the choice of tactics for taking profits and limiting losses. Before deciding to part with the found or developed strategy due to the low results of MO, you should pay attention to the ratio of stops and takes.

A small size of loss limitation leads to an increase in the number of negative transactions and the accumulation of losses. If a trader trades the EUR/USD pair intraday, he must take into account that the “trading noise” is on average 30 points and will often trigger stop losses located in this zone.

A take/stop ratio of 2 to 1 increases the expected value. It is believed that takes and stops should not be below parity (1 to 1).

A decrease in the number of transactions can lead to an increase in the value of MO. Traders use time filters, trading during the session in areas that coincide in time with the work of the stock exchanges of the countries to which the pair’s currencies belong.

Improving the quality of entries - purchases or sales of currency pairs. Filters are introduced into the trading system to allow transactions at significant points. These are historical highs and lows, candles that coincide in trend on lower and higher timeframes, indicator readings with a long period (from 50), etc.

Features of mathematical expectation when scalping

Scalping is characterized by a large number of intraday trades with a low positive MO value. The small size of stops in this case is an exception, justified by high trading activity. With a slight prevalence of profit over loss, earnings come from a large number of intraday trades.

There are no exceptions to the remaining tactical rules - the scalper applies a fixed take value that exceeds the stop level. The search for the optimal value of the expected value is achieved by selecting the time for holding the transaction; the scalper should not “sit out” or work when there is no volatility.

The parameter under consideration does not alone determine the feasibility of adopting a strategy. Performance assessment is based on a comprehensive analysis of test results.