2022 Alibaba global mathematics competition, question 4, huhushengwei (blind box problem, truck problem) solution ideas

subject

come from 2022 Alibaba global mathematics competition The first 4 topic （ Single topic selection ）

Basic concepts

Mathematical expectation

In probability and Statistics , Mathematical expectation (mathematic expectation)（ Or the average , Also called expectation ） Is the probability of each possible result in the test multiplied by the sum of its results , Is one of the most basic mathematical characteristics .

In this question , The mathematical expectation is the average number of cards that need to be drawn in order to be collected .

If fat tiger wants to win, he will sleep , The probability of winning is 1/10 Words , The relationship between the number of times required and its probability , It is geometrically distributed （ All the previous failures , The last success , Geometric distribution ）, Probability average 10 Is the average number of times fat tigers need .（10 The last game won , The probability of this situation is about 0.04, But it is the average probability of the whole probability diagram , Corresponding 10 That's the average number ）

Satisfy the problem of geometric distribution , From the calculation, there will be a simple formula —— Expectation is the reciprocal of probability .

Fat tiger 10% The probability of success , Of course, the expectation is 10.

Answer key

“ Tiger Shengwei ” problem

Mathematical expectation = 1（ Draw any card for the first time ）+ 3/2（ Draw a non duplicate card for the second time ）+ 3（ Draw a non repeated card for the third time ） = 5.5

“ Water margin 108 card ” problem

It is also the multiplication of mathematical expectations

After mathematical derivation, the expectation is calculated ：568（ On average, you need to buy 568 package ）

You can see ,n The bigger it is , The greater the mathematical expectation .

“ A tiger makes a mighty tiger ” problem

There is “ repeat ” The situation of .
This problem can be solved by Monte Carlo method

Monte Carlo method is also called statistical simulation method 、 Random sampling technique , It's a random simulation method , A calculation method based on probability and statistical theory , Is to use random numbers ( Or more common pseudo-random numbers ) To solve a lot of computational problems . Connect the problem to a certain probability model , Statistical simulation or sampling with an electronic computer , To get an approximate solution to the problem . To symbolically show the probabilistic statistical characteristics of this method , Therefore, it is named after casino Monte Carlo .

Algorithm is as follows ：（ Many simulations , Every time, cycle the card drawing until you get a tiger , Record times , Finally calculate the average number ）

Considering the time , A more scientific method is needed to solve .
Let's have a classified discussion .
You need to draw four cards , So we classify all kinds of cards , There will be four rounds in each case , Which round represents the number of cards drawn , Calculate the number of times needed （ That is the expectation of this round ）.

	First round expectation	Second round expectation	The third round of expectations	Fourth round expectation
The first round draws tiger	1（ The tiger ）	1（ at will ）	3/2	3
The first round draws to live , The second round draws tiger	1（ raw ）	1（ The tiger ）	3/2	3
The first round draws to live , The second round draws Wei	1（ raw ）	1（ Wei ）	3	3
The first round draws to live , The second round draws students	1（ raw ）	1（ raw ）	/	/
In the first round, we won , The second round draws tiger	1（ Wei ）	1（ The tiger ）	3/2	3
In the first round, we won , The second round draws Wei	1（ Wei ）	1（ Wei ）	/	/
In the first round, we won , The second round draws students	1（ Wei ）	1（ raw ）	3	3

In the table , Both the first round and the second round are 1 Why , It's because the situation is separated , Which one we draw is determined by us , So the probability is 1, The expectation is 1.
/ It means that this situation is special , Because drawing to life is equivalent to returning to the origin .
It seems impossible to calculate , In fact, we only need to set the situation of the first round of drawing to students and the second round of drawing to students as x, Then we can calculate x What are your mathematical expectations .
x（ The first round draws to live ） = 1/3（ The second round draws tiger ） * 9/2 + 1/3（ The second round draws Wei ） * 6 + 1/3 * x（ The second round draws students ）+ 1（ Once in the second round ）

notes ： In fact, it can also be understood as ：
x（ The first round draws to live ） = 1/3（ The second round draws tiger ） * 11/2 + 1/3（ The second round draws Wei ） * 7 + 1/3 * (x+1)（ The second round draws students ）

The resulting x yes 27/4
Empathy , You can also get the situation of winning for the first time 27/4
Then the table becomes ：

The final mathematical expectation is to multiply the three cases , be equal to 7.333.

Reference resources ：
【 Ali Shusai 】 How difficult is the single choice question ？ You were chosen by Mathematics , To answer ！