Odds of Winning the Lottery Using the Same Numbers Repeatedly Better/Worse?

Does the probability of winning the lottery differ between randomly generated numbers vs. selecting the same numbers every time?

Specifically. I'm interested in a breakdown of the odds per number for a given set of numbers that comprise a single US Powerball drawing (five white numbers plus the one powerball number), and how they arrive at the odds seen here: http://www.powerball.com/powerball/pb_prizes.asp

Tying that back to my original question, I was interested if playing the same numbers every drawing changes those odds.

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No. Why would it? –  Qiaochu Yuan Jan 6 '11 at 16:42
I'm sorry if that came across as rude; I was trying to ask a Socratic question. But I don't see how replying with an even ruder remark is at all productive. –  Qiaochu Yuan Jan 6 '11 at 16:51
I see you're a moderator. I've asked this question on other exclusive forums and had people thumb their noses at me. Stack exchange sites and all the mechanics therein are supposed to help eliminate that kind of unhelpful answer. You used the comment system to answer without explaining, thus avoiding the very system that makes a stack exchange Q and A better than phpBB or vBulletin forums. Excuse me if I'm less than impressed. I'm not a statistician, but I'm interested in learning. If a break down of odds were given in an answer that could help me understand why, then I would up-vote you. –  afilbert Jan 6 '11 at 17:02
@afilbert: I am not thumbing my nose at you, and I apologize if it appears that way. I would like it if you could explain to me a plausible reason for the answer to this question to be "yes," and then I can write an answer which addresses your specific thought process. If you don't want to do that, that's fine. –  Qiaochu Yuan Jan 6 '11 at 17:08
I think I'm interested in seeing what the odds are per number, and how those odds are arrived at overall. For instance, the odds of actually winning the jackpot are ridiculous (1 in 195,249,054.00). I'm stupid when it comes to how they arrive at figures like that, which is also why I posed the question. I posed the question here hoping to start a list of answers that would be vying for superiority by going into deeper detail. Perhaps I should have clarified that I was looking for a breakdown of the odds per number and a more detailed answer than simply 'yes' or 'no.' –  afilbert Jan 6 '11 at 17:18

So here is how Powerball works. You choose five different numbers between $1$ and $59$ inclusive (the white balls) and one number between $1$ and $39$ inclusive (the red ball). If the white balls match the winning numbers for the white balls, in any order, and if the red ball matches the winning number for the red ball, then you win the jackpot.

Because you can match the white balls in any order, the Powerball winning numbers are usually presented from smallest to largest, so if you order your numbers from smallest to largest, the two sequences have to match. The number of ways to pick five different numbers in any order from $1$ to $59$ is

$$\frac{59 \cdot 58 \cdot 57 \cdot 56 \cdot 55}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 5006386$$

and every choice of five different numbers in increasing order has the same probability (one over the above number) of being chosen. One way to get the above number is as follows: first, pretend that order matters. Then there are $59$ possibilities for the first number. Since there are $58$ possibilities left, there are $58$ possibilities for the second number. And so forth. But since order doesn't matter, you can draw any set of five numbers in $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ different ways (five factorial), so you have to divide by that.

Matching the red ball is easy: there are $39$ choices, so you have a $\frac{1}{39}$ chance of doing it. So, in summary, your odds of winning the jackpot from any choice of numbers is

$$\frac{1}{5006386 \cdot 39} = \frac{1}{195249054}$$

just as reported on the Powerball website. In other words, it's one over the total number of possible tickets. (The other probabilities reported on the website are slightly harder to calculate, but not by much; for that you need to learn about something called the inclusion-exclusion principle.)

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Maybe now is a good time to fully explain my comments above. Suppose that the answer to the question were "yes," and that your odds were either better if you used the same number again or worse. Unless the lottery is keeping specific tabs on what numbers you pick (which, again, would be very strange), this effect would have to happen to everybody. Now imagine that every week you and your neighbor switches what numbers you pick. Then either both of your odds have improved, or both of your odds have gotten worse. But this is nonsense, since this case is indistinguishable from the case... –  Qiaochu Yuan Jan 6 '11 at 18:01
...where you picked the same numbers you always pick. So however the lottery works, as long as it works the same way for everybody, it can't discriminate between whether you chose the same numbers as last week or not. So if you (the generic you, not you specifically) think that choosing the same numbers has an effect on your odds of winning, you have to postulate some mechanism for the lottery to keep track of these numbers. And in my comments above I was trying to tease out whether you believed such a mechanism could exist, or get you to realize that it shouldn't. –  Qiaochu Yuan Jan 6 '11 at 18:03
Understood, and thank you for the detail! I've heard from many of my friends and family that if you play your own pet numbers over and over, somehow you have a greater chance of winning. Intuitively I was skeptical but couldn't express mathematically why I felt that way. My intention was to have somebody well versed in mathematics better articulate that intuition. –  afilbert Jan 6 '11 at 20:07
@afilbert: I guess that's a form of the Gambler's fallacy: en.wikipedia.org/wiki/Gambler's_fallacy –  Qiaochu Yuan Jan 6 '11 at 21:08
@afilbert It's a bad idea to play pet numbers. Pet numbers are numbers that are special to you for some reason. The set of numbers that are special to people is much smaller than the set of all possible numbers. (Eg. people often pick dates which means 19 comes up often in people's pet numbers.) The net effect is that you are more likely to choose the same combination as someone else and have to share the prize. Pick randomly instead. And don't just pick numbers as people often pick the same 'random' numbers (eg. 37 is popular). –  Dan Piponi Jan 18 '11 at 22:31

The probability of winning does not depend on the specific numbers selected and the numbers drawn for each lottery drawing have no dependence on previous drawings, so there is no benefit to playing the same numbers every time.

For most U.S. state lotteries, it is beneficial to choose numbers above 31 when possible as many people play numbers based on dates and so picking numbers above 31 lowers the likelihood of split pots. But, really, playing the lottery is a losing proposition regardless of the mechanics.

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Exactly, it's the math tax. –  uncle brad Jan 6 '11 at 17:08
I guess you meant to say "there is no advantage or disadvantage to playing the same numbers each time." –  Qiaochu Yuan Jan 6 '11 at 17:12
Tried to up-vote you Isaac, and thanks for the response (need a few more rep to up-vote). I was debating with Qiaochu Yuan and made my question a moving target, for that I apologize. He answered with the kind of greater detail I was looking for. I'll come back around and up-vote you once I can. You can count on it. –  afilbert Jan 6 '11 at 17:52
The lottery is nearly always a losing proposition, but not necessarily always, and it does not always have the same expected return. Although the chances of winning with a random ticket remain the same, the payoff does not. In addition, the chances of having to share the prize depend on the number of ticket sold. –  phv3773 Jul 13 '11 at 14:54
@phv3773: While it's true that there are times when the expected value (ignoring split prizes) of a ticket is a net gain, the probability of winning is so small that it might as well be 0, putting the expected value back to a net loss. Millions of people will play in such a case, and each will have a positive expected value, but essentially all of them will lose money. –  Isaac Jul 13 '11 at 15:59

No. Since the lottery numbers are randomly generated each time independently of the previous draws, the probability of winning is always the same.

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No, unless you know the exact probability that each number will come out; in that case, you should always choose the most probable numbers.

Another case in which you should choose the same numbers if you have some extra information is in Isaac's answer, but that only applies if you change your goal to maximize your gain (or rather, minimize your loss) rather than maximize the probability of winning.

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Does the probability differ number to number? I've heard that in a pair of dice, it's more probable that some combinations will get rolled as opposed to others. So in a range of 1 through 59, are there certain numbers that are bound to be chosen more often? –  afilbert Jan 8 '11 at 0:06
That's a question for the company which runs the Lottery. I'd bet they make an effort for the distribution to be as uniform as possible. –  Yuval Filmus Jan 8 '11 at 6:53

I have nothing to add to Qiaochu's and Isaac's answers and comments on the probability of winning. If you are buying tickets to maximize the earnings if you happen to win then the selection of numbers becomes important. It is best to try to pick numbers that other people are unlikely to pick. Examine the numbers when no one wins or only one person wins and try to find a property that many of these numbers have in common. If you find such a property then examine numbers when there were multiple winners and see if few of these numbers satisfy the property. Some years ago, when the rules were somewhat different than today I noticed that there were very few winners when the numbers contained consecutive integers. I do not know if this is still a good method for choosing numbers that fewer other people choose.

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If you really want to maximize earnings, you should only play lotteries with actual flaws that you can exploit (see for example wired.com/magazine/2011/01/ff_lottery/all/1 ). –  Qiaochu Yuan Jul 14 '11 at 2:44

surely what needs to be considered in this situation is not just the probability of playing one game but if you set out to play the same numbers for a set period of time. For example using the same set of numbers for 52 weeks.

Rather than trying to win the lottery in one week, you are looking at the best way to guess what lottery numbers may come up over the year, in which case playing the same numbers throughout the year would increase probability.

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Unfortunately this is wrong ("playing the same numbers throughout the year would increase probability"), assuming the drawings are done fairly (i.e. independent, uniform distributions of outcomes). It's also a bit late to post, at least without having read the accepted answer and making an explicit effort to address some point where your answer differs. –  hardmath Jul 13 '11 at 14:25

protected by Qiaochu YuanJul 14 '11 at 2:39

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