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1. Considering the 3 digits from 000 to 999, if 3 identical digits appear, the score can be obtained. What is the average score of these 1,000 numbers? (EX: 111 scores 1 point, 999 scores 9 points)
2. Continuation 1. If 0 can replace other numbers, what will the total average become? (EX:000 can be regarded as 999 to get 9 points)
3. Consider the 5 digits from 00000 to 99999, from left to right consecutively appearing more than 3 of the same number can get their corresponding connection scores, NNNXX can get 3N points, NNNNX can get 4N points, and NNNNN can get 5N points. 0 can replace other numbers. What is the total average of these 100,000 numbers?
Before solving the problem, explain to the reader the connection between these three questions and the slot machine model. If we replace the numbers 1~9 with the pay symbol that appears in the game, and 0 with the wild symbol, then these three questions can be regarded as three slot games.
For the first two questions, readers who are familiar with permutations and combinations should be able to get answers easily. The third question is a little bit more complicated, because there are more cases to be considered, and the amount of calculation will be larger.
Let me explain to you first that when calculating the model of the slot machine, the probability engineer will adjust the model by creating a spreadsheet. The spreadsheet is actually storing the calculation process in an Excel file. The following will demonstrate how to use Excel to create a spreadsheet.
In addition my purpose is to import the model thinking of the slot machine, I will solve the problem from this perspective. Therefore the method given may not be the fastest way to solve these three problems. Readers are also asked to focus on the thinking of solving the problem. This will be more helpful for subsequent learning.
I use I0~I9 to represent the numbers 0-9, and enter the number of times each digit appears in the table on the left, and then calculate how many digits there are in each digit, and how many three-digits combination in total.
In the right part, the number combination that can get the score is listed, and then the number of occurrences of each score combination is calculated. After the score is multiplied, the total score and the total average can be obtained.
Here's why we need to use the table on the left, because when the number of occurrences of each number is not equal (EX: the hundred digits have 3 I1/I3/I4, 2 I6), just change the weight of the table. The data is updated, and the results of this calculation are displayed almost simultaneously.
Followed by the method of the second question, use the form of the first question to expand.
Here is a little explanation of the method of the second question, because 0 can replace other numbers, so the way to get 3 I1 will become I0 or I1. It can be represented by I01 below, and there will be a total of 2*2*2=8 In this way, it also includes I0-I0-I0 where will be specially scored. One reason is that its score is usually different from others, and the other reason is that it is convenient for other score combinations. Deduct the number of points it has scored.
Followed by the practice of the third question, the second question table is expanded.
First X0 represents a number that will not allow 0 to be continuous (1-9 in total 9), X01 represents a number that will not allow I1 to be continuous (2 to 9 in total 8), and then the part of the score combination is 5-digits to 3-digits in sequence, please note the scores of I0-I0-I0-I0-X0. At first glance, it seems to be a set of 4 consecutive scores of I0, but because I0 can replace any number, so regardless of the single digit Which number N appears in X0 will make this group of numbers form a 5-of-a-kind score of N. Therefore it is necessary to compare N's 5-of-a-kind score or I0's 4-of-a-kind score for I1~I9, and deduct the number of repetitions of low scores. This is what the newly added table on the far right is doing.
Note: The reel strips used in the par sheet may not be the real strips of the game.
Parsheet_01 - Basic Concept
Game Instructions
Before starting to learn the slot machine model, there are a few math questions, please read and think about it:1. Considering the 3 digits from 000 to 999, if 3 identical digits appear, the score can be obtained. What is the average score of these 1,000 numbers? (EX: 111 scores 1 point, 999 scores 9 points)
2. Continuation 1. If 0 can replace other numbers, what will the total average become? (EX:000 can be regarded as 999 to get 9 points)
3. Consider the 5 digits from 00000 to 99999, from left to right consecutively appearing more than 3 of the same number can get their corresponding connection scores, NNNXX can get 3N points, NNNNX can get 4N points, and NNNNN can get 5N points. 0 can replace other numbers. What is the total average of these 100,000 numbers?
Before solving the problem, explain to the reader the connection between these three questions and the slot machine model. If we replace the numbers 1~9 with the pay symbol that appears in the game, and 0 with the wild symbol, then these three questions can be regarded as three slot games.
For the first two questions, readers who are familiar with permutations and combinations should be able to get answers easily. The third question is a little bit more complicated, because there are more cases to be considered, and the amount of calculation will be larger.
Let me explain to you first that when calculating the model of the slot machine, the probability engineer will adjust the model by creating a spreadsheet. The spreadsheet is actually storing the calculation process in an Excel file. The following will demonstrate how to use Excel to create a spreadsheet.
In addition my purpose is to import the model thinking of the slot machine, I will solve the problem from this perspective. Therefore the method given may not be the fastest way to solve these three problems. Readers are also asked to focus on the thinking of solving the problem. This will be more helpful for subsequent learning.
Model Descriptions
Game Calculation
I use I0~I9 to represent the numbers 0-9, and enter the number of times each digit appears in the table on the left, and then calculate how many digits there are in each digit, and how many three-digits combination in total.
In the right part, the number combination that can get the score is listed, and then the number of occurrences of each score combination is calculated. After the score is multiplied, the total score and the total average can be obtained.
Here's why we need to use the table on the left, because when the number of occurrences of each number is not equal (EX: the hundred digits have 3 I1/I3/I4, 2 I6), just change the weight of the table. The data is updated, and the results of this calculation are displayed almost simultaneously.
Followed by the method of the second question, use the form of the first question to expand.
Here is a little explanation of the method of the second question, because 0 can replace other numbers, so the way to get 3 I1 will become I0 or I1. It can be represented by I01 below, and there will be a total of 2*2*2=8 In this way, it also includes I0-I0-I0 where will be specially scored. One reason is that its score is usually different from others, and the other reason is that it is convenient for other score combinations. Deduct the number of points it has scored.
Followed by the practice of the third question, the second question table is expanded.
First X0 represents a number that will not allow 0 to be continuous (1-9 in total 9), X01 represents a number that will not allow I1 to be continuous (2 to 9 in total 8), and then the part of the score combination is 5-digits to 3-digits in sequence, please note the scores of I0-I0-I0-I0-X0. At first glance, it seems to be a set of 4 consecutive scores of I0, but because I0 can replace any number, so regardless of the single digit Which number N appears in X0 will make this group of numbers form a 5-of-a-kind score of N. Therefore it is necessary to compare N's 5-of-a-kind score or I0's 4-of-a-kind score for I1~I9, and deduct the number of repetitions of low scores. This is what the newly added table on the far right is doing.
Simulation Result
File Download
Here is the par sheet of this game, if you are interested, you can download the par sheet file for research.Note: The reel strips used in the par sheet may not be the real strips of the game.
Parsheet_01 - Basic Concept
由模型怎么得出模拟结果的
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