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pcbouhid
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A personal lotto game
« on: Jun 16th, 2009, 7:55am » |
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48 members of our social group use the weekly results of our national lotto draw for our competition.
Each person selects a set of 5 numbers from just one of the four grids of the map shown below, such that they are all in a row (e.g. row 15: 92, 13, 36, 24 and 79) or in a column (e.g. column 28: 75, 42, 14, 99 and 60) or diagonally (e.g. diagonal 37: 32, 65, 93, 76 and 10).
Our national lotto draws, each week, 5 numbers from the set of 100 numbers (00-99), and we permanently mark these 5 numbers in our map. The winner in our competition will be the person who firstly has his 5 numbers marked.
What is the probability that our competition:
a) ends in the first week?
b) ends in the second week?
c) ends in the third week?
And the most important:
d) What is the expected number of weeks for us to have a winner?
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| « Last Edit: Jun 16th, 2009, 9:42am by pcbouhid » |
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towr
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Re: A personal lotto game
« Reply #1 on: Jun 16th, 2009, 8:42am » |
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on Jun 16th, 2009, 7:55am, pcbouhid wrote:| (Iīm not succeeding in attaching the grid example) |
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Do you get an error?
You could always put it on imageshack, and just insert it with [img] tags.
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pcbouhid
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Re: A personal lotto game
« Reply #2 on: Jun 16th, 2009, 8:53am » |
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Hi towr, good to "see" you again.
When submiting the problem, I clicked the attached file, selected the .jpeg image, but itīs not loaded.
Iīve been away for a long time. What is "imageshack"? I know how to insert the tags.
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towr
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Re: A personal lotto game
« Reply #3 on: Jun 16th, 2009, 9:18am » |
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on Jun 16th, 2009, 8:53am, pcbouhid wrote:| Hi towr, good to "see" you again. |
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Yeah, you too, it's been a while, hasn't it.
Quote:When submiting the problem, I clicked the attached file, selected the .jpeg image, but itīs not loaded.
Iīve been away for a long time. |
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Did you preview before posting, because that removes the attachment. Aside from that if anything goes wrong, it should give an error as far as I know.
Quote:| What is "imageshack"? I know how to insert the tags. |
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It's a free image host, http://www.imageshack.us/
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pcbouhid
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Re: A personal lotto game
« Reply #4 on: Jun 16th, 2009, 9:44am » |
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Ty, towr, I used "imageshack". But it seems not too visible. What do you think?
Edited to say that I realized that clicking in it, it enlarges. Ty... and Iīm waiting some probabilities, that it seems to me, will be evaluated with a computer program.
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| « Last Edit: Jun 16th, 2009, 9:46am by pcbouhid » |
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towr
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Re: A personal lotto game
« Reply #5 on: Jun 16th, 2009, 10:28am » |
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Does everyone pick a different set of 5 numbers? i.e. are you basically asking how long before one of the rows/columns/diagonals is filled in?
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pcbouhid
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Re: A personal lotto game
« Reply #6 on: Jun 16th, 2009, 5:15pm » |
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Yes, towr. We are altogether 48 friends, each one with a set of 5 numbers. The numbers outside the grid identify each one of us. But, as you can see, one number is shared by 2 or 3 (if it is in a diagonal) persons. Clear?
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towr
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Re: A personal lotto game
« Reply #7 on: Jun 17th, 2009, 1:23am » |
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Experimental results, a simulation of 1,000,000 trials:
2, 158, 1354, 5186, 13413, 27022, 46190, 68631, 90756, 109891, 120713, 120513, 109923, 91615, 71121, 49880, 32202, 19601, 10866, 5778, 2840, 1343, 602, 230, 104, 42, 14, 7, 3
expected number of weeks until there's a winner: 11.728
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| « Last Edit: Jun 17th, 2009, 3:32am by towr » |
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Ronno
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Re: A personal lotto game
« Reply #8 on: Jun 17th, 2009, 3:54am » |
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What is the sequence? If they are the number of weeks then how do they average to 11.728?
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towr
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Re: A personal lotto game
« Reply #9 on: Jun 17th, 2009, 4:15am » |
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on Jun 17th, 2009, 3:54am, Ronno wrote:| What is the sequence? If they are the number of weeks then how do they average to 11.728? |
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The sequence is the number of trials that end in that week out of the total1,000,000 trials.
The expected number of weeks is the weighted average, i.e. the sum of week-number times proportion of trials ending that week.
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pcbouhid
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Re: A personal lotto game
« Reply #10 on: Jun 17th, 2009, 6:50am » |
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Towr, the numbers you got are very much close to the numbers a friend (in another site) got with his program, which are: 0 103 1123 4791 12950 25175 42652 63851 87470 101577 116655 119701 110794 92946 75606 54705 38340 23298 13551 7667 3834 1853 790 335 131 59 14 14 15 trials that took 1 through 29 weeks respectively. There were no cases requiring over 29 weeks. This made the expected number of weeks 11.95692..
I became surprised with the expected number of weeks to end the competition (12 weeks... 3 months) because I didnīt recall a competition that lasted too long. We do this a long time, each one pays something like US$20 and the winner (or winners) receives US$960. Not bad return - if you are the winner, of course -, donīt you think?
Ty for your great help. Cheers.
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towr
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Re: A personal lotto game
« Reply #11 on: Jun 17th, 2009, 8:00am » |
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Hmmm, after a million trials, I'd think your friend's number and my numbers should be closer. The error should be in the order of 1/100th (=10/ #trials), not 1/4th
The results seem to be skewed with respect to eachother, mostly; where I have higher results for early weeks, and lower results for later weeks. If I had to guess, I might suspect he didn't pull 5 distinct numbers for each lotto draw, but rather drew them with replacement. And that suspicion is confirmed by introducing that same error in my program, which then gives 11.976 (for 100k trials).
But I suppose if the accuracy is good enough for your purposes, it doesn't really matter.
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pcbouhid
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Re: A personal lotto game
« Reply #12 on: Jun 17th, 2009, 11:16am » |
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I accessed the forum to post the same comment, raised by him ("the results should be much closer"). Since we use the official results of a federal loto draw, the solution must consider "replacement" (every week is a "new game" for the people in general). Obviously that for my purpose, the number 12 found by you two fills my need.
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SMQ
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Re: A personal lotto game
« Reply #13 on: Jun 17th, 2009, 11:29am » |
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But could they draw, e.g. "13, 28, 56, 77, 13" for a single week's results? I think that's was towr is saying: that your friend's program appears to be allowing that possibility while towr's program guarantees the five numbers for a single drawing are distinct.
--SMQ
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towr
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Re: A personal lotto game
« Reply #14 on: Jun 17th, 2009, 1:02pm » |
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on Jun 17th, 2009, 11:29am, SMQ wrote:| But could they draw, e.g. "13, 28, 56, 77, 13" for a single week's results? I think that's was towr is saying: that your friend's program appears to be allowing that possibility while towr's program guarantees the five numbers for a single drawing are distinct. |
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That is indeed precisely what I'm saying.
The numbers in a given week have to be distinct, but those of different weeks may overlap.
So (keeping the order of the balls), there are 100!/95! ways to draw the balls for a given week, and not 1005; a difference of 10% in valid draws.
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pcbouhid
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Re: A personal lotto game
« Reply #15 on: Jun 17th, 2009, 2:24pm » |
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Sorry, I misunderstood your question, and I was wondering: if there are no replacement how can towr found results for more than 20 weeks?
Ok. As I said, itīs a federal lottery here in Brazil, so the 5 numbers drawn, each week, are all different.
Iīll ask my friend about his drawings, but I think this point is clear to him. Probably I made myself clearer to him than here.
Btw, isnīt 100!/(95!*5!)?
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towr
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Re: A personal lotto game
« Reply #16 on: Jun 17th, 2009, 2:44pm » |
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on Jun 17th, 2009, 2:24pm, pcbouhid wrote:| Btw, isnīt 100!/(95!*5!)? |
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That's without order. This makes it easier to compare (since removing order from the 1005 case is a bit more difficult)
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pcbouhid
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Re: A personal lotto game
« Reply #17 on: Jun 17th, 2009, 3:07pm » |
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Yes, youīre right.
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Re: A personal lotto game
« Reply #18 on: Jun 18th, 2009, 4:11am » |
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It might interest you to know that this isn't really a fair game. The people that choose the diagonals have a smaller probability to win.
Well, ok, technically they have the same probability to win, but when they win chances are greater that they have to share that win with someone else. And assuming the prize money is shared amongst the winners, this means they get less on average. And it's quite noticeable, around 6-7%.
[edit]The middle row/column of each square also seems to have an ever so slight smaller expected payoff; but the difference is much less noticeable (about 0.2%).[/edit]
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| « Last Edit: Jun 18th, 2009, 5:36am by towr » |
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pcbouhid
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Re: A personal lotto game
« Reply #19 on: Jul 1st, 2009, 8:54am » |
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Thank you one more time, towr. I had that feeling about diagonals (each number is shared by 3 people, and the others by 2 only) but without a computer itīs hard to find the percentages you found.
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