Lotto Probability

[Rodney]

While there is an equal probability of any single number being chosen, there are some differences when you look at the probability for the numbers as a group. The result is a different gambler's fallacy, but fun to ponder.

The WA State lotto is a 6 pick system from numbers 1-49 and there is no special ball and numbers cannot repeat.

Take the number 1 (it also works for #1-5 and #45-49 respectively). While the chances of getting #1 should be equal to any other number, 1 appears less likely to be chosen because it can only occur in positions #1-44. #7 OTOH, has 5 more chances of being chosen since it can go in the positions of #1-49.

So if you take the odds of picking any of the 49 numbers (ping pong balls are used) each number should have an equal chance of being chosen. But if you calculate out the chance of each of the 6 positions individually, you get different odds. If you then add the odds of each position together, #1-5 and #45-49 should occur less frequently.

The reason it is a gambler's fallacy is while #1 cannot end up in position #45-49, it occurs more often in positions #1-#6 and vice versa for #45-49.

You can buy a group of numbers that increase your odds. In a 6 pick system, you can buy all the combinations for 7 numbers pretty inexpensively. When you start getting all the combinations for 8 or more the cost goes up pretty fast. I try to play the 7 number group even though it only increases the odds of winning by a tiny amount. But hey, that's at least a real increase and not a fallacious one.

The number frequency charts are another means of increasing probability. They really only have meaning in a physical system where there are true variables in the system such as bouncing and bumping balls. If it is a computer generated random number, then you'd have to show something in the program was not truly random. Otherwise you'd just be seeing a coincidental difference in the number frequency and not a true variation.

I don't follow your logic. In a 6-pick system with 49 numbers, the odds of any of the 49 numbers coming up in a given drawing should be identical: 6 chances in 49. Similarly, the odds of any combination of 6 numbers coming up in a given drawing should be identical: (6/49) * (5/48) * (4/47) * (3/46) * (2/45) * (1/44) = 1/13,983,816.

 

[macgyver]

Take the number 1 (it also works for #1-5 and #45-49 respectively). While the chances of getting #1 should be equal to any other number, 1 appears less likely to be chosen because it can only occur in positions #1-44. #7 OTOH, has 5 more chances of being chosen since it can go in the positions of #1-49.

You'll need to explain this better, I have no idea what you mean. I'm with Rodney: the likelyhood of any number appearing should be identical.

I understand that combinations reappearing when looked at as a group may have some additional connotations, but there's no reason, for instance, that 1,2,3,4,5,6 can't be the winning numbers two weeks in a row. It's just as likely to happen each time the numbers are drawn.

 

[RandFan]

While there is an equal probability of any single number being chosen, there are some differences when you look at the probability for the numbers as a group. The result is a different gambler's fallacy, but fun to ponder.

The WA State lotto is a 6 pick system from numbers 1-49 and there is no special ball and numbers cannot repeat.

Take the number 1 (it also works for #1-5 and #45-49 respectively). While the chances of getting #1 should be equal to any other number, 1 appears less likely to be chosen because it can only occur in positions #1-44. #7 OTOH, has 5 more chances of being chosen since it can go in the positions of #1-49.

So if you take the odds of picking any of the 49 numbers (ping pong balls are used) each number should have an equal chance of being chosen. But if you calculate out the chance of each of the 6 positions individually, you get different odds. If you then add the odds of each position together, #1-5 and #45-49 should occur less frequently.

The reason it is a gambler's fallacy is while #1 cannot end up in position #45-49, it occurs more often in positions #1-#6 and vice versa for #45-49.

You can buy a group of numbers that increase your odds. In a 6 pick system, you can buy all the combinations for 7 numbers pretty inexpensively. When you start getting all the combinations for 8 or more the cost goes up pretty fast. I try to play the 7 number group even though it only increases the odds of winning by a tiny amount. But hey, that's at least a real increase and not a fallacious one.

The number frequency charts are another means of increasing probability. They really only have meaning in a physical system where there are true variables in the system such as bouncing and bumping balls. If it is a computer generated random number, then you'd have to show something in the program was not truly random. Otherwise you'd just be seeing a coincidental difference in the number frequency and not a true variation.

Yeah, I'm not getting this either. Could be I'm just dumb though. It has happened before.

 

[macgyver]

Yeah, I'm not getting this either. Could be I'm just dumb though. It has happened before.

Well, at least you're prolific - Mr. Penultimate Amazing!

 

[Skeptic Ginger]

I can't believe you guys don't understand my post. Not enough time spent contemplating the Universe and how to win at lotto at 3 am I guess. I found all sorts of ways to look at the lotto probability and this one was one of the easiest.

So let's try again. Hopefully you didn't all cheat and look at the spoiler since it ruins all the fun.

Yes yes yes, Rodney, if you calculate the odds of getting any single number, you get the same odds for each.

But take another approach. Calculate the odds of getting any number for EACH of the six number places individually. The first number is in place #1, the second number is in place #2 the third number is in place #3 and so on.

You cannot get #49 in place #1 because you have to have 5 smaller numbers to the left of #49. You cannot have #1 in place #6 because you have to have 5 numbers to the right of #1. When you calculate the odds for the number in position #2, the chances of getting # 49 in that position is zero. If you calculate the odds of getting #1 in the second to last place, the odds are zero, in third to last place the odds are zero and so on.

So the numbers 1 and 49 can only go in 1 each of the 6 number places in the lotto. If you calculate the odds one at a time of getting numbers 1-49 in each of the six places then add those odds together, numbers 6 through 45 have greater odds because those numbers can come up 6 times while #1 and #49 for example, can only come up once.

And yes, it is a fallacy. I said that already.

 

[RandFan]

Well, at least you're prolific - Mr. Penultimate Amazing!

I try.

 

[Rasmus]

And yes, it is a fallacy. I said that already.

It is not a fallacy, it is simply quite wrong.

The numbers are drawn randomly, so nothing stops #49 to be drawn first. (Not in any of the lotteries I know ,at least, and also not in the game you described in your post.)

What rules I then use to display the numbers is a different issue, and it matters not to the nature of the game. In principle, there is nothing to stop me from displaying the numbers in random order.

None of this alters anyone's chances of winning, as the order in which the numbers are drawn doesn't matter. (And if it did matter, there would times 180 as many possibility, yet the 49 could still be drawn first.)

 

[BobK]

While there is an equal probability of any single number being chosen, there are some differences when you look at the probability for the numbers as a group. The result is a different gambler's fallacy, but fun to ponder.

The WA State lotto is a 6 pick system from numbers 1-49 and there is no special ball and numbers cannot repeat.

Take the number 1 (it also works for #1-5 and #45-49 respectively). While the chances of getting #1 should be equal to any other number, 1 appears less likely to be chosen because it can only occur in positions #1-44. #7 OTOH, has 5 more chances of being chosen since it can go in the positions of #1-49.

So if you take the odds of picking any of the 49 numbers (ping pong balls are used) each number should have an equal chance of being chosen. But if you calculate out the chance of each of the 6 positions individually, you get different odds. If you then add the odds of each position together, #1-5 and #45-49 should occur less frequently.

The reason it is a gambler's fallacy is while #1 cannot end up in position #45-49, it occurs more often in positions #1-#6 and vice versa for #45-49.

In the lotto the numbers have no value. They are the names of objects. As such, any selected ordering of them is arbitrary. Value order is no more valid than aphabetical order or drawn order. I fail to see how the spoiler applies.

You can buy a group of numbers that increase your odds. In a 6 pick system, you can buy all the combinations for 7 numbers pretty inexpensively. When you start getting all the combinations for 8 or more the cost goes up pretty fast. I try to play the 7 number group even though it only increases the odds of winning by a tiny amount. But hey, that's at least a real increase and not a fallacious one.

Wow! You spend a lot on the lottery.

You increase your odds of winning, but you also increase your investment proportionally. You might win in 100 years instead of 500. The value of each individual bets stays the same.

The number frequency charts are another means of increasing probability. They really only have meaning in a physical system where there are true variables in the system such as bouncing and bumping balls. If it is a computer generated random number, then you'd have to show something in the program was not truly random. Otherwise you'd just be seeing a coincidental difference in the number frequency and not a true variation.

Forget the frequency charts. Take a ball weighted in such a fashion that it comes up twice as often as it should. In 48 pick 6 lottery, that ball will on average come up 1 time out of 4. At 2 drawings per week, it will be a long time before you could have any confidence you found an anomaly. If it becomes obvious to others(which it will) they will also do the same as you, and with more tickets on the number the pool is more likely to get divided. The people running the lottery will also notice. Don't you think they might be watching out for the same anomalies you are?

If you want to do it, do it for entertainment. Just don't fool yourself into thinking you're doing anything productive to improve your chance of winning.

 

[LW]

I wasn't aware that statistics are available regarding how often number combinations are played in any lotteries. Do you have a link to any such statistics?

Complete statistics are not available. Those figures that I posted come from the info page of Veikkaus Oy, the company who runs the Finnish lotto. In addition to those tidbits, they mention that the most often played single numbers are: 9, 3, 21, 10, and 8 and the least often played is 39.

That is pretty much the extent of the statistics available.

 

[kevin]

As near as I can tell the Washington lottery is a match the numbers only lotto, not a match the numbers AND match the position. They always display the results in increasing order to make matching against your card easier.

The cute little demo they have shows the numbers coming out in any order:
http://www.walottery.com/PopUp/gameLotto.htm

They have some statistics on which numbers come up most frequently here:

http://www.walottery.com/sections/LotteryGames/Lotto.aspx?Page=FDN

the #1 has popped up 203 times while 7 has only shown up 175.

 

[Ladewig]

If I understand Skeptigirl's post, she is suggesting that after the six numbers are drawn they are placed in ascending order. Newspapers always publish the results in ascending order rather than the order drawn because lotto ticket machines print the numbers in ascending order. She was pointing out that innumerate people sometimes calculate probability by looking at groups of numbers that are ranked in ascending order. That is why #49 cannot appear in the first position: there must be five numbers smaller than #49 in any particular drawing.

 

[Rasmus]

If I understand Skeptigirl's post, she is suggesting that after the six numbers are drawn they are placed in ascending order. Newspapers always publish the results in ascending order rather than the order drawn because lotto ticket machines print the numbers in ascending order. She was pointing out that innumerate people sometimes calculate probability by looking at groups of numbers that are ranked in ascending order. That is why #49 cannot appear in the first position: there must be five numbers smaller than #49 in any particular drawing.

I have never come across anybody who was seriously making that kind of argument.

You would have to ignorant about the way the numbers are drawn, and assume a very elaborate drawing scheme instead. Basically, it would only work if the numbers available for the next turn would be decided after each previous number.

In that case, however, we would see 44 - 45 - 46 - 47 - 48 - 49 as a result very often! Everytime number 44 would be drawn first, the end result would be set!

I truly cannot imagine that anybody would think that a specific number combination would come up about once a year ...

 

[Skeptic Ginger]

Like I said, I guess you guys aren't prone to contemplating the Universe and making millions winning the lotto at 3 am. Haven't you ever taken a math or probability problem and played around with it, looking at it from different angles? All I did was calculate each of the 6 choices one at a time instead of calculating the odds of the next number coming up. You can take lists of winning numbers. Put them in columns so each row has 6 cells. Graph out all the numbers for column one, column two, etc. It has very interesting results. While the lowest number that can be in column 6 is 6, results in that column are much more likely to be >20. And in the first column results are much more likely to be <20.

I know it is an illusion. I've said that 3 times now. So I don't understand why all the claims are being made here of how stupid the idea is. It's as if you all think I'm saying it affects the real probability of the numbers. I'm only showing you a particular phenomenon that turns up when you look at how often numbers are in each position, rather than how often any single number comes up.

If you just look at how many times each number has been chosen, you can see right away that the method of looking at the columns is flawed. But think how it looks at 3 am. It looks like you should space your choices out and avoid numbers <6 and >44.

I'm sorry I brought it up.

 

[69dodge]

Haven't you ever taken a math or probability problem and played around with it, looking at it from different angles?

Definitely. That's totally the right thing to do. I don't know why people are giving you such a hard time about it. I thought it was amusing.

It reminded me a bit of this: Consider (the circumference of) a circle and suppose you pick three points on it at random. What can you say about the lengths of the three arcs that the points divide it into? That is, how often (i.e., with what probabilities) do the various possible lengths occur?

Here it's interesting to look at the distribution of the shortest arc or the longest one or the middle one, similar to the way you looked at the distribution of the smallest chosen lotto number or the largest or the one in some other position.

If three points is too easy, next try four.

Or five.

Or arbitrary n. (That should keep you busy for a while. . .)

 

[kevin]

Like I said, I guess you guys aren't prone to contemplating the Universe and making millions winning the lotto at 3 am.

only when beer is involved. which usually makes my already poor math skills really bad, although my universe contemplating capability usually soars.

 

[Bob Klase]

An interesting side note- possibly related to this thread because it shows the use of a fallacy in promoting the lottery.

A few years ago Florida changed it's lotto. Instead of 1 to 49, the numbers available are 1 to 53. A week or so before the change took effect I saw (a lot of) ads on TV promoting the change as a good thing- "Florida's Lottery- now even more numbers to pick from!". Obviously the state knew that a significant number of people were dumb enough to believe that having more numbers to pick from was something to be desired.

 

[tsg]

Is there any less chance that the same group of 6 numbers will be randomly generated more than once?

We went through this a while ago when a newspaper erroneously reported that the chances of the same pick 3 numbers coming up twice in a row are a million to one.

Picking 3 specific numbers and having them win tonight and tomorrow: one in a million.

Any 3 number combination winning tonight and tomorrow: one in a thousand (there's a thousand more ways it can happen).

Any 3 number combiation winning two days in a row eventually: pretty much certain. Assuming a game every day, you can expect it to happen roughly once every three years.

 

[macgyver]

I'm sorry I brought it up.

Actually, until your last post, I still had no idea what you are talking about. I think I was confused by your talking about the number's "position", since I've always considered that the numbers would be drawn completely randomly and only displayed in ascending order for ease of use.

Now I get what you mean, and I see that it's an interesting thought experiment.

Another way to describe the same thing (and just as erroneous) would be the idea that each successive number picked has slightly better odds of being picked than the last. That's because each number cannot repeat, so the first number is 1 in 49, the second is 1 in 48 the third is 1 in 47 etc...

But I'm sure this is also an illusion...

 

[Skeptic Ginger]

An interesting side note- possibly related to this thread because it shows the use of a fallacy in promoting the lottery.

A few years ago Florida changed it's lotto. Instead of 1 to 49, the numbers available are 1 to 53. A week or so before the change took effect I saw (a lot of) ads on TV promoting the change as a good thing- "Florida's Lottery- now even more numbers to pick from!". Obviously the state knew that a significant number of people were dumb enough to believe that having more numbers to pick from was something to be desired.

That's amazing. And sad as well. They've gone up and down in this state as well because making it harder to win means bigger jackpots and more people get lotto fever when the pots are big. I wonder how much is media hype about the big pots (free advertising) and how much is the psychology of it. I imagine a bit of both.

 

[Skeptic Ginger]

Actually, until your last post, I still had no idea what you are talking about. I think I was confused by your talking about the number's "position", since I've always considered that the numbers would be drawn completely randomly and only displayed in ascending order for ease of use.

Now I get what you mean, and I see that it's an interesting thought experiment.

Another way to describe the same thing (and just as erroneous) would be the idea that each successive number picked has slightly better odds of being picked than the last. That's because each number cannot repeat, so the first number is 1 in 49, the second is 1 in 48 the third is 1 in 47 etc...

But I'm sure this is also an illusion...

But how can you use that to pick numbers? Once they start picking numbers, bets are closed.

The second number is now one in 48, then 1 in 47 and so on so the probability does go up. The funny part is, then the least probable (the first number picked), is always picked giving it 100% odds yet it had lower odds in the beginning. I'd say this is correct probability per number picked but useless as it has no impact on the previous number picked. In other words, the individual number probability changes but the probability relationship between each subsequent number is unaffected. Another 3 am contemplation subject nonetheless.