Random picking of 3 digit numbers

[7Kardz]

If the odds of randomly picking a 3 digit number correctly are 1000-1, then what are the odds of that same number being picked 4 times in 364 picks?

 

[MarkFL]

Hello and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Okay, that being said...I think what we need here is the following:

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Binomial Probability Formula​

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\(\displaystyle P(x)={n \choose x}p^x(1-p)^{n-x}\)​

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Can you identify $x,\,n,\,p$?

 

[7Kardz]

Thanks for the quick reply. Unfortunately it seems i misunderstood what this site does for people, like me that is. I haven't taken a math course since grade 12, which was 20 years ago. So, i couldn't even tell you what those letters represent in that formula you showed me. Maybe if they were switched out with numbers i might have a chance at figuring it out, but it's "greek to me" as they say. So, again, sorry i misunderstood mathehelpboards.com and wasted your time. I've spent the last 10 years in university and college, graduating from both with a BA degree and diploma, but it was in music and music/audio production, so no math involved beyond basic math, and at 39 years old i don't have the time to go back and learn algebra or whatever those letters are, and I'm sure you don't have the time to teach me either. I am truly sorry for posting my question.

 

[MarkFL]

Thanks for the quick reply. Unfortunately it seems i misunderstood what this site does for people, like me that is. I haven't taken a math course since grade 12, which was 20 years ago. So, i couldn't even tell you what those letters represent in that formula you showed me. Maybe if they were switched out with numbers i might have a chance at figuring it out, but it's "greek to me" as they say. So, again, sorry i misunderstood mathehelpboards.com and wasted your time. I've spent the last 10 years in university and college, graduating from both with a BA degree and diploma, but it was in music and music/audio production, so no math involved beyond basic math, and at 39 years old i don't have the time to go back and learn algebra or whatever those letters are, and I'm sure you don't have the time to teach me either. I am truly sorry for posting my question.

Hello again,

We primarily cater to students (whether being formally educated or self-learning) of mathematics, and our goal is to help provide guidance and direction in solving problems with which our members are having difficulty...so we generally try to offer nudges in the right direction, in the hope that the student will do the heavy lifting and gain an understanding by taking part in the process of solving the problem. Our main goal is not so much to provide answers to questions, but to help students learn how to solve problems effectively.

Since your question does not arise from some kind of participation in a statistics/probability course, may I ask where the question comes from and why you are interested in the solution? I will be happy to answer your question, but please indulge me first by letting me know the origin of the problem. :D

 

[MarkFL]

I'm not sure, but I think the OP replied to the thread subscription email rather than replying here in the thread...this is what I received:

Wow, thanks. I wasn’t expecting that response; I appreciate it. You’re correct, I'm not in a math course, or even took much math in secondary and post secondary studies, but I've always been interested in stats and probabilities, mostly in relation to games such as poker and other gambling games. I should say right off the bat that in no way am I looking for “an edge” or the answer to some question to win some game.

My recent interest in probabilities came from wanting to investigate the possibility of suspicious activity within the daily Pick 3 game that our government lottery puts out. Here’s what I'm trying to figure out and why: how probable is it that the same 3 digit number would come up 4 times within a year, or 364 (no draw Christmas Day) days. That’s the first question..then: how probable is it the someone would win twice, picking one set of numbers which come up twice within 25 draws/days, and/or they picked 4 sets of 3 and 2 were correct meaning they won twice within 25 days. I'm sorry I can’t express this more mathematically. The game basically works like this: on each ticket players can pick 1-4 sets of 3 digit numbers, pick how many draws they want to enter, up to 25 (per ticket) and then choose how much to wager, \$1,\$2, \$5, or \$10. The payout is 500-1 so \$5000 is the top prize, but recently, in the “recent winner’s” picture section on the lottery website, there’s a guy holding a sign for the Pick 3 game and he’s holding a sign for \$10,000. When I emailed them to ask “How did this guy win \$10K when the top prize is \$5k?” Their response was “he must’ve won twice on the same ticket; we pay out accumulative wins”. That seems so improbable to me that I decided to research a bit, and discovered that there have been several numbers come up multiple times within short periods: 212 and 341 came up FOUR times just in the last 12 months, but the number I've played for 12 months, 801, last came up almost 4 1/2 years or 1640 odd draws ago.

The system that pick these numbers for the game randomly is computerized, so I thought “is it possible that it could be manipulated somehow to choose certain numbers programmed into it by cheaters, who then bet the maximum amount on their tickets to win double the max prize? By the way, in order to for this person, the one who won the \$10,000, to have won just by chance, and for him to improve his odds as much as possible, he would’ve had to have bought at least a \$250 lottery ticket (Pick 3): 25 draws at \$10 each, or up to a \$1000 ticket: 4 picks at \$10 each for 25 draws. Have you ever noticed anyone buying a single lottery ticket for \$250 up to \$1000? In 20 years I haven’t, but for this guy to have had the best chance to pull off two maximum prize wins on a single ticket, that’s what it would’ve taken. Of course he could've play less draws on the ticket, but that makes this whole thing even more improbable, at least to me. So, to begin with, I decided to see about the math involved in all this so next email I send the lottery corp asking them to find out how he won that amount, I can at least point out the odds of all this and hopefully someone there will look into it. Yes, I'm being a “sore loser” in the sense that I'm not happy that 801 hasn’t come up in 4 1/2 years, and I've been playing it for 1 year, but I only play \$1 per draw and can only ever hope to win \$500; it’s not about the money for me. \$500 isn’t much; it’s about putting the time into something you feel will happen, then it doesn’t, or doesn’t yet, and in the meantime you discover some other person had won double the top prize and the lottery corp. can’t even give you a straight answer as to how.

I know this has nothing to do with your website, but I appreciate you reading this much already. I don’t expect you to put time into figuring these things out, and I totally understand; I feel better for being honest about why I posted my question in the first place, and to have got my ideas out of my head and into this email. Thanks again for responding and please, don’t worry about solving this, I really just wanted to be honest about why I asked in the first place. I do really enjoy math because of the absoluteness of it, you know? There’s not to many certainties in life but 1+1 will always be 2 and I love that. Have a great day, and...that’s that I guess.

I do want to answer all of your questions, but at the moment I don't have time. I thought I would post your reply for our other staff/helpers to see though for now. I will post a reply within 24 hours, although anyone else with thoughts on the questions should feel free to post.

 

[MarkFL]

Okay, first let's find the probability that the same randomly generated 3 digit number (000-999) would occur 4 times in 364 trials:

\(\displaystyle P(X)={364 \choose 4}\left(\frac{1}{1000}\right)^4\left(\frac{999}{1000}\right)^{360}\approx0.000501866608196667039600409441627399489862168794293813245\)

So, we should expect this to happen about once every two thousand years. :)

Now, for the probability that the same number would come up twice within a 25 day period, we find:

\(\displaystyle P(X)={25 \choose 2}\left(\frac{1}{1000}\right)^2\left(\frac{999}{1000}\right)^{23}\approx0.000293175371346435510699749385814581880425209979316825992\)

This is even less likely than the first scenario.

 

[squidsk]

I'll just comment on one thing since this is a common misconception amount people without a math or computing background.

The system that pick these numbers for the game randomly is computerized

There is no real random number generation done by computers. The number picked are pseudo-random and are generated by a mathematical formula. Some of the math formulas used provide better random numbers than the others. Regardless of which formulas are being used if you know the seed, or number used to start the calculations, you would know the order of the numbers, also if you know the formula being used but do not know the seed it is likely that you could increase of your odds of successfully guessing future random numbers, especially with lotteries since you have access to all past numbers chosen by the psuedo-random number generator. So if a weak pseudo-random number generator was chosen it is very possible someone could be manipulating the system.