Complex Mystery Number Probability Question

Can you clarify the question? When you say "on the list of 10 numbers", are you saying that ten numbers are chosen from the pool with replacement? What is the pool, the numbers between 1 and 15?

Yes, to clarify - the list of 10 numbers is: 4, 8, 6, 6, 3, 3, 10, 3, 10, 4. This list was randomly generated from a 1 to 15 pool, with repeats allowed. Now, these numbers were generated by 10 people who were asked, what was their favorite number between 1 and 15. The person who asked them, would blindly draw a number and that would be the mystery number. That statistics were developed over 5 years and then analyzed. Which brings us to this discussion. I hope that helps.

Sorry, failing to comprehend here.

- The mystery number is a single number.
- The list of ten numbers is fixed.
- Sometimes it appears and sometimes it doesn't.

???

It sounds like the mystery number is changing or the list of 10 numbers is changing over multiple pulls and that you are aggregating the results. Which one is it? Are you saying a new set of 10 numbers is pulled from the ten you have there with replacement?

All that said and with a complete lack of understanding on my part, I think all you need is the following two statements.

20% of the time, a number that is greater by 1 than the mystery number, does not appear 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.

This means that 80% of the time, a number that is greater by 1 than the mystery number appears. Since the numbers on the list are 3, 4, 6, 8 and 10, the only possible numbers are 2, 3, 5, 7 and 9. But from the second entry, the only possible numbers are 1, 2, 4, 6 and 8. Only the number 2 overlaps from those two groups.

Some additional 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. In my initial post, I provided the statistics we came up with and the latest 10 numbers selected out of a pool of 15 choices.

Hence, what I am looking for is the steps to calculate the probability of each number being the mystery number - which ultimately, the highest result should be mystery number.

In this case, I am personally selecting 3 as the possible mystery number. This is why I would like to understand the steps to identify the probability of 3. The steps should be the same for each number. Based on the statistics provided, 3 appears three times - if it is the mystery number, then we start off with 10%. We then have 4, which is greater than 3 by 1 and which appears twice on the list of 10, so we now have 25%. Lastly, 5 is greater than 3 by 2 and is not on the list, so we have 20%. This is where I don't know how to calculate the probability that 3 might be the mystery number. (.1, .25, .2).

Does that make sense?

Honestly, no. The mystery number is changing with each query of ten people, so there is no one mystery number and the mystery number random between 1 and 15. In order to use statistics in the way you want, you need the events to be related somehow. If they had the same MN, you could use the results of drawings 1-100 to predict what would happen on drawing 101. In your case, every single drawing is different including have a different MN. Just an example, if the MN was 15, then the probability of a number being one or two greater than 15 is zero. You know that happens sometimes and it just gets lumped into the overall results. But you can't look at the overall results and say the MN is never 15. You know that it is.

So here's a better question. Same data set, but you know that the MN's are randomly distributed between 1 and 15. What is the list of ten numbers?

All I need here is a way to calculate what the mystery number based on the information we have. It does not matter if this works or not - this is something I wanted to basically try. The questions you are asking are going deep into the weeds, meaning that it does not truly matter if the the mystery number was a constant each pull or not. The fact that the percentages are what they are, is all I am interested in.

I am positive that there is a way to calculate this based on the information provided. I just don't understand the theorems well enough to do this on my own, hence why I am asking. Additionally, my breakdown is clear. So let's try this again.

We asked 10 folks, to pick a number between 1 and 15. The results were, 4, 8, 6, 6, 3, 3, 10, 3, 10, 4. Meaning that the first person picked 4, the second picked 8, the third picked 6, and so on. Next, my friend picked a number between 1 and 15 out of a bag. The number he picked, is the mystery number.

Now that is established, let's move on to the other times we have done this. What I described above, has been something we have done during our D&D sessions for nearly 5 years now. We have recorded those choices and the mystery number each time. Recently, we decided to see what the stats on those events were. What we came up with was interesting and we then researched via Google regarding how to develop a mystery number question based on what we had. We figured out that we needed to calculate the probability of each number based on what we knew.

So, based on what I have already provided, I am not sure where the confusion is. Further, if you cannot assist, that does not mean that someone else cannot. I will wait to see if there is someone else. Given that I know that certain percentages need to be multiplied against each other and then divided etc. That is all I am looking for.

I asked my friend and the way that the information was analyzed was based on something he did not properly articulate because he thought it would be confusing. When he analyzed the information, all numbers ranging from 1 to 7 fell under the +1 and +2 rules. Then all numbers ranging from 8 to 15 fell under a different rule, -1 and -2. I explained that is not how the breakdown is presented and that it would be impossible to determine if 14 and 15 were the MN.

I hope that helps. I am trying to patiently navigate this question while dealing with a mom who is in the hospital. So my apologies.

Resolved on my own.