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In a pick 4 lottery, one winning number is chosen between 0-9999.

So the probability of picking the correct number is 1/10000. And the probability of not picking the correct number is 9999/10000.

So after 15000 draws, we have the historical data of the past 15000 winning numbers. Analyzing the 15000 numbers, the number 1234 has never been the winning number before.

Am I correct to say the probability of the number 1234 not being drawn after 15,000 draws is (9999/10000)^15000 = 22.3% ? So the probability of the number 1234 being drawn out after 15000 is 77.7% ?

So after 30,000 draws, the probability of the number 1234 not being drawn is (9999/10000)^30000 = 4.9% ?

I get as the number of draws increase, the probability of the number 1234 not being drawn decreases.

Would that mean as the number of draws increases, the number 1234 should have an even higher probability to come out ?

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    $\begingroup$ Nope. Each draw is independent. Look up "gambler's fallacy." $\endgroup$ – Cheerful Parsnip Sep 20 '18 at 16:11
  • $\begingroup$ Thanks, its helpful. I understanding more about the reasoning behind the gambler's fallacy now. But what I just don't understand is - does that means after 15000 draws the probability of the number being drawn out is 77%. So does that mean if I bought any number, I had a 77% chance of winning ? $\endgroup$ – Frankie139 Sep 20 '18 at 16:41
  • $\begingroup$ Yes, obviously the probability of 1234 occuring sometime during $30,000$ draws is higher than the probability of 1234 occuring sometime during $15,000$ draws. But the probability that 1234 occuring sometime during the draws 15,001 to 30,000 is not any higher than the probability occuring sometime durin the draws 1 to 15,000. And there is nothing in you argument to imply it would be. $\endgroup$ – fleablood Sep 20 '18 at 16:43
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Your calculations are correct but note that at $30,000$ draws the chance that it has not been drawn is about $4.9\%$ or about $1$ in $20$. As there are $10,000$ numbers, there should be about $490$ numbers that have not been drawn yet. The fact that $1234$ or any other number has not been drawn is very weak evidence that the draws are not random.

Lotteries are supposed to work very hard to make the draw truly uniformly random. If the draw is truly uniformly random, there is no information in the history that is of use. It could be that there is intentional bias because somebody who runs the lottery tips off a friend so the friend can have a higher probability of winning. Deleting one or a few numbers does not improve the friend's chances much and I would imagine they vary the bias to avoid getting caught. There could be an unintentional bias. If there are $10,000$ balls to draw from the one with number $1234$ might be too heavy and settle to the bottom of the pile. You could detect that with history, but you would need a lot more data to make a good case.

Just don't play the lottery and you don't have to worry about it.

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  • $\begingroup$ Thanks, I'm starting to understand what you're trying to say. But I'm just a bit confused still. So after 30,000 draws there should be roughly 490 numbers that has not been drawn out yet. Why isn't it a strategy to buy those 490 numbers that hasn't been drawn out yet ? Because after say 40,000 draws, the probability of a number not being drawn is 1.8%. So shouldnt a majority of those 490 numbers come up between 30,000 and 40,000 draws? $\endgroup$ – Frankie139 Sep 20 '18 at 16:47
  • $\begingroup$ It is a strategy to buy those balls, but no better a strategy than buying the others. There is no reason to think they are more likely to come up than the ones that have come up. $\endgroup$ – Ross Millikan Nov 6 '18 at 15:11
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You definitely should not choose 1234 because lots of other people are likely to choose it as well. As a result, if you do miraculously win, you get to split your earning with lots of other folks. Thus, the expected value of that pick is less than that of a truly random pick.

Similarly, you should not pick 4321, 1111, or any number with an obvious pattern. You should also not pick the date - or any date for that matter.

Of course, none of this has any bearing on winning the lottery but it can affect the amount you win, if you do win. Specifically, independence implies that as the number of draws increase, the probability of the number 1234 not being drawn next remains constant.

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If the lottery has done their homework (i.e. made sure the drawing is completely unbiased), then your choice of number should not affect the probability of winning at all (it may, however, affect the expectation value of the won amount, because the more people chose a certain number, the less money each single winner gets if it is drawn; since many people falsely believe seldom-drawn numbers are more likely, it's then a good idea to avoid them).

It the lottery has not done their homework, and there's a chance that the drawing is consistently biased, then you should definitely not take a number that hasn't yet been drawn, because if there are numbers that are more likely to be drawn, it is more likely that they already have been drawn, and therefore you're more likely to bet on such a number if you bet on a number that already has been drawn (indeed, your best bet would be the number that has been drawn most). Of course if you factor in the amount of money you expect to win, that analysis will likely be thwarted by the number of other people doing it, too; unless the lottery is blatantly biased, your best bet will be to choose “unremarkable” numbers.

Of course when looking at the expected outcome, the absolute best bet is not to play at all. It's the only strategy with non-negative expectation value.

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