# Modeling: Most Probable Drawn Numbers

Please forgive me for any grammar mistake.

I'm a civil engineer and completed my Msc (Maths) focusing on Numerical Study 10 years ago. After my semi retirement as a result of my financial freedom, i have been studying some practical Maths problem for fun.

Recently I've been trying to model and solve a 2 digit lottery drawing game, and i failed. It's purely my imagination since i didn't see this in anywhere. But who knows it may exist?

Suppose we have a lottery game of 2 digits, drawn from 2 separate but identical electrical drums as lottery company always have. Each drum consists of 10 balls, numbered from 0 to 9, to be drawn as a pair and the drawn balls are to be replaced. In one game, 12 pairs of numbers to be drawn as winning numbers, on every Saturday and Sunday.

Eg A particular Saturday: 09, 21, 04, 31, 48, 61, 00, 32, 99, 98, 11, 99 Sunday: another 12 pairs of numbers

My question is: if you have the result of last 1000 game, how do you calculate the most probable drawn numbers (one or two pairs) for the next drawing?

Any idea?

• Uhh.... "the most probable drawn numbers for the next drawing..." Why do you suppose such a thing would exist? Have you ever heard of the Gambler's Fallacy? – JMoravitz Jun 14 '19 at 15:18
• Not clear what you are asking. If the numbers are chosen uniformly at random, independently of each other, then each number is equally likely to be drawn, regardless of the history. Do you suspect bias in one or the other selection? Well, what evidence have you got for that? I'd plot out the incidence of each digit to confirm that the distribution is plausibly uniform. Take it from there. – lulu Jun 14 '19 at 15:19
• @howardpotts Something I want to highlight given the recent responses. I notice that you say there are identical electric drums and that the pulls are independent. You don't note that the balls are equally likely to be drawn. Are we to assume that for the purposes of the problem not all of the balls are created equal? – Kitter Catter Jun 14 '19 at 15:27

There is no most probable number, because the events are independent and all the outcomes equally likely.

In other words, even if you had a million outcomes to analyze, the probability of drawing a particular number -- say, $$42$$ -- is $$0.01$$, just like for every other number you'd care to choose.

A short anecdote:

One of my teachers in high school was talking about the "pick 3" lottery. He looked at the previous results, and would draw patterns of the likelihood of having two digits the same in the number ($$343$$, for example).

I (respectfully) explained to him that there was nothing to be gained from looking at these patterns, and that it wouldn't help.

The following week, he actually won \\$400 at the pick 3 because he hit the number, and it had a repeated digit in it. I honestly can't remember whether I let him think he was right or not (haha). The truth is that he had the same $$0.1\%$$ chance he would have any other day.

Say that we have a good lottery that keeps their balls in tip top shape to avoid accusations of fraud. In this case any kind of analysis will yield spurious results since the probability of drawing any ball is a-priori known to be independent and each ball is equally likely.

Say that you believed that underlying this lottery some balls were more likely than others. As others have noted you should still test out your idea before putting money behind it. You have 12000 draws so you should be able to establish your hypothesis by doing some statistical testing, many good solutions here: How many rolls do I need to determine if my dice are fair?

With the volume of data you may be able to get away with some Monte Carlo simulations directly from the data you collected. This might also help with trying to do some optimization/viability calculations for whatever you end up doing with the results.

I wanted to add that if you follow a bunch of ideas of imbalance rather than just in the digits you should be careful with how you establish credibility of your results since you are increasing the likelihood that you discover a spurious result.