Complex Mystery Number Probability Question

[furthark]

How can I solve the problem? I am looking for the calculation steps, using the number 3 as an example. We don't have to go through each number since the steps would be the same for each.

Some background. For approximately 5 years, a friend and I would run this game which we would ask ten folks, independently, to pick their favorite number between 1 and 15. Separately, a mystery number was selected by drawing it out of a bag which contained 15 balls, each labeled 1 through 15. We would record the results and one day we decided to see what the statistics were. We are now stuck - we want to identify how those statistics can be used to solve for the mystery number. Below you will find the statistics we came up with and at the end, you will find the 10 numbers that were recently selected out of a pool of 15 choices.

Solve for the mystery number.

40% of the time, the mystery number does not appear on the list of 10 numbers.
35% of the time, the mystery number appears once on the list of 10 numbers.
15% of the time, the mystery number appears twice on the list of 10 numbers.
10% of the time, the mystery number appears three times on the list of 10 numbers.

20% of the time, a number that is greater by 1 than the mystery number, does not appear on the list of 10 numbers.
25% of the time, a number that is greater by 1 than the mystery number, appears once on the list of 10 numbers.
30% of the time, a number that is greater by 1 than the mystery number, appears twice on the list of 10 numbers.
15% of the time, a number that is greater by 1 than the mystery number, appears three times on the list of 10 numbers.
10% of the time, a number that is greater by 1 than the mystery number, appears four times on the list of 10 numbers.

20% of the time, a number that is greater by 2 than the mystery number, does not appear on the list of 10 numbers.
30% of the time, a number that is greater by 2 than the mystery number, appears once on the list of 10 numbers.
25% of the time, a number that is greater by 2 than the mystery number, appears twice on the list of 10 numbers.
15% of the time, a number that is greater by 2 than the mystery number, appears three times on the list of 10 numbers.
10% of the time, a number that is greater by 2 than the mystery number, appears four times on the list of 10 numbers.

This is the list of 10 numbers: 4, 8, 6, 6, 3, 3, 10, 3, 10, 4. The selection pool was 1 through 15.

 

[Steven G]

Separately, a mystery number was selected by drawing it out of a bag which contained 15 balls, each labeled 1 through 15.

Here is where you lose me. If each ball is labeled 1 through 15, then how do you tell the difference between each ball?

 

[ImMAD]

It seems to me that you are trying in a rather convoluted way to estimate the mystery number, which is a random number between 1 and 15 and completely independent of the 10 numbers that your friends picked, by using the information of the numbers that your friends picked. That will not give you accurate predictions, and if the data points you to some specific mystery number, then it just means your sample is to small.

 

[furthark]

Here is where you lose me. If each ball is labeled 1 through 15, then how do you tell the difference between each ball?

Stuck on semantics, a bit silly and childish, but I understand. So, let me help you get past your point of confusion and reword that sentence so that you are not lost: Separately, a mystery number was selected by drawing it out of a bag containing 15 balls labeled with numbers ranging from 1 to 15.

I hope that helps get you back on track.

If your position here is to simply troll posts, then you have served your purpose. If you can help, please do. If you cannot, then thanks for the chuckle.

 

[furthark]

It seems to me that you are trying in a rather convoluted way to estimate the mystery number, which is a random number between 1 and 15 and completely independent of the 10 numbers that your friends picked, by using the information of the numbers that your friends picked. That will not give you accurate predictions, and if the data points you to some specific mystery number, then it just means your sample is to small.

Here's a breakdown of the process and the sample was not small, it was established over 5 years.

This all started during D&D game nights, approximately 5 years ago. The game lead, the DM, would draw a random number between 1 and 15, and the players would pick a number between 1 and 15. The purpose was to provide a system of random luck for the players who selected the number or who were within a few points from the random pick. Luckily, the DM wrote down those results down and he recently analyzed those results. The whole purpose with analyzing them was to see if we could incorporate something far more unique to the game. The results of his analysis are provided in the breakdown I provided in the initial post. That is the background.

Now, via another forum - I had to clarify that my friend had not shared with me how he did his analysis and embarrassingly, I had to elaborate on that. Unfortunately, that was identified after I had posted in this forum and was distracted with semantics.

Here's how he did the analysis of the data he collected over those 5 years, which is not properly reflected on the breakdown I provided.

All player selected numbers that ranged between 1 and 7 were calculated as x = x, x = x+1, and x = x+2. Then, all numbers ranging between 8 and 15, were calculated as x = x, x = x -1, and x = x - 2. Hence, the breakdown is a bit deceiving and since I cannot edit the post, and deleting and re-posting is considered a form of spam, this meant that I was stuck with what I posted. I am using this post to clarify your question and shed additional clarity on the post itself. Further, I asked Chat GPT and the breakdown started to lead me in a positive direction, but I would like to still like to see what a mathematician's breakdown on this.

I hope this helps.

 

[furthark]

Resolved on my own.

 

[ImMAD]

If I understand you correctly, a player chooses a number, and then the DM draws a random number, and compares. Then, event X occurs for players within for example 2 of the mystery number, and event Y otherwise. The obvious downside of this would be that players that choose a number near the edges would have a low probability of event X occuring, and players near 7.5 a high probability. Then, do you circumvent this by saying that for example, if the mystery number is 1, and we want all players with +- 2 to be accounted for in the event, that you count the numbers 14 and 15 of being 2 away from the mystery number? In that case, all players with number {1,2,3,14,15} will have event X occur, and {4,5,6,7,8,9,10,11,12,13} will have event Y occur. Do I understand this correctly?

Next, what is the unknown quantity that you wish to estimate, and with what purpose? At the moment, I think you want to see if there are certain numbers you can pick that are more close to the mystery number on average, and if you can devise an optimal strategy based on that. This might be incorrect from my part, hence me asking you to state this very precisely, so that we can help you better with your question.

Btw, I would not advice chatGPT for statistical questions, in my opinion it still makes too many mistakes on this topic.

 

[furthark]

If I understand you correctly, a player chooses a number, and then the DM draws a random number, and compares. Then, event X occurs for players within for example 2 of the mystery number, and event Y otherwise. The obvious downside of this would be that players that choose a number near the edges would have a low probability of event X occurring, and players near 7.5 a high probability. Then, do you circumvent this by saying that for example, if the mystery number is 1, and we want all players with +- 2 to be accounted for in the event, that you count the numbers 14 and 15 of being 2 away from the mystery number? In that case, all players with number {1,2,3,14,15} will have event X occur, and {4,5,6,7,8,9,10,11,12,13} will have event Y occur. Do I understand this correctly?

Next, what is the unknown quantity that you wish to estimate, and with what purpose? At the moment, I think you want to see if there are certain numbers you can pick that are more close to the mystery number on average, and if you can devise an optimal strategy based on that. This might be incorrect from my part, hence me asking you to state this very precisely, so that we can help you better with your question.

Btw, I would not advice ChatGPT for statistical questions, in my opinion it still makes too many mistakes on this topic.

The solution was to use Bayes' Theorem:
[math]P(A\mid B)=\frac {P(B\mid A) \cdot P(A)}{P(B)}[/math]
Where:
A, B = events
P(A|B)=probability of A given B is true
P(B|A)=probability of B given A is true
P(A), P(B)= the independent probabilities of A and B

Hence, in my case, which I wanted to know if 3 was the number drawn:
A = Event that 3 is the drawn number
B = Event that the observed list of numbers is [4, 8, 6, 6, 3, 3, 10, 3, 10, 4]

Then we have to answer P(B|A):
P(B|A) = P(3 occurrences of x | 3 is drawn) * P(2 occurrences of x+1 | 4 is drawn) * P(0 occurrences of x+2 | 5 is drawn)

My error was not understanding how P(B) would have to be answered, I then realized that I would have to calculate each P(B|A):
[math]P(B) = Σ [P(B|Ai) * P(Ai)] for i = 1 to 15[/math]
Btw, all numbers ranging between 1 and 7 were calculated as:
P(B|A) = P(Number of occurrences of x | i is drawn) * P(Number of occurrences of x+1 | i+1 is drawn) * P(Number of occurrences of x+2 | i+2 is drawn)
Example: i=3, i+1 = 4, i+2 = 5, etc.

All numbers ranging between 8 and 15 were calculated as:
P(B|A) = P(Number of occurrences of x | i is drawn) * P(Number of occurrences of x-1 | i-1 is drawn) * P(Number of occurrences of x-2 | i-2 is drawn)
Example: i=14, i-1 = 13, i-2 = 12, etc.

After that, all I had to do is establish P(A), 1/15.

That is how I resolved this. If I made any errors, then I would like to know where I made those errors.

Lastly, ChatGPT was used to nudge me in a direction. I fully understand that it is not reliable. However, it is a decent search engine when it comes to identifying the general context of what you are looking for.

The ultimate answer to the problem ended up being that 5 and 11 have the highest probability of being the drawn number, with the next most likely choices being numbers 7, 9, and 12. The mystery number that was drawn in this case was 5. So, for this attempt it worked just fine - understanding that complete randomness cannot be predicted.

That's all this was meant to be in the end. A learning exercise and so many across four separate forums made this out to be almost sacrilegious or even taboo. I was basically trolled and for what? To prove that mathematicians cannot relate with other humans or that their value is greater than those who are not part of their community? A bit silly and childish at the end of the day.

ImMAD, thanks for input and please let me know if my breakdown makes sense and if it is accurate. Thank you very much and Happy Easter to you and yours.