It should be noted that the expected value is not necessarily equal to the common sense "expectation"-"expected value" is not necessarily equal to every result. The expected value is the average of the output values of variables. The expected value is not necessarily contained in the set of output values of variables.

The law of large numbers stipulates that as the number of repetitions approaches infinity, the arithmetic average of numerical values almost inevitably converges to the expected value.

Extended data:

Lottery problem

Suppose a department store has a batch of daily products that are about to expire and need urgent treatment. The supermarket owner designed a free lottery to deal with these products. Carton contains 20 balls of the same size, 10 balls 10 points, 10 balls 5 points. Among them, 10 balls are drawn, and the sum of the scores of 10 balls is the winning point. The awards are as follows:

First prize 100 points, a freezer is worth 2500 yuan;

50 points for the second prize and one TV set, worth 1000 yuan;

95 points for the third prize and 8 bottles of shampoo, worth 178 yuan;

55 points for the fourth prize and 4 bottles of shampoo, worth 88 yuan;

60 points for the fifth prize and 2 bottles of shampoo, worth 44 yuan;

Sixth prize of 65 points, a box of toothpaste, worth 8 yuan;

70 points for the seventh prize and a bag of washing powder, worth 5 yuan;

85 points in the eighth prize, a bar of soap, worth 3 yuan;

90 points for the ninth prize, one toothbrush, worth 2 yuan;

75 points and 80 points of the tenth prize are divided into excellent prizes. Only 22 yuan, the cost price, will get a bottle of shampoo;

Analysis: On the surface, the whole activity is beneficial to customers. The first prize to the ninth prize are all in vain, and only the tenth prize receives a little cost price. But after analysis, can you know that the business is really losing money? Can customers really get the chance to win the grand prize from it? Get its expectation and the truth will come out.

Only 65,438+065,438+0 found the score of 65,438+00 balls. X is used to indicate the winning times of lottery winners, and E (x) is calculated as-10.098, which means that the average merchant will get 10.098 yuan for each lottery, and each lottery winner will spend an average of 65,438 yuan.

So it can be seen that customers are really standing on bargains? On the contrary, merchants can not only get rid of expired goods, but also gather a lot of popularity for supermarkets in this way, which is multifaceted.

The boss of this department store used mathematical expectation to estimate that he would not lose money, so he won the free lottery. In the end, he won more than one move, from which we can see the importance of mathematical expectation as a scientific method in economic decision-making.

Sports competition issues:

Table tennis is our national game. In the last century, ice hockey also brought some diplomacy to China. China has an absolute advantage in this sport. Now I want to ask a question about the arrangement of table tennis matches:

Suppose Germany (German star Bohr also has many fans in China) plays China. There are two competition systems, one is three players per team, and the other is five players per team. Which is more beneficial to China?

Analysis: Because of the advantages of China team in this competition, let's assume that the winning percentage of each German player in China team is 60%, and then we only need to compare the corresponding mathematical expectations of the two teams.

Baidu Encyclopedia-Mathematical Expectation