Winning proximity

[nimzov]

Suppose there is a mega-huge-lottery where they draw 6 numbers from 100.
The winning combination came out and is:

12 - 14 - 30 - 46 - 52 - 68

Suppose also you have 5 tickets for this lottery with the following combination:

Ticket 1 : 11 - 13 - 29 - 45 - 51 - 67
Ticket 2 : 21 - 41 - 03 - 64 - 25 - 86
Ticket 3 : 06 - 07 - 15 - 23 - 26 - 34
Ticket 4 : 07 - 27 - 39 - 69 - 90 - 95
Ticket 5 : 12 - 55 - 61 - 83 - 85 - 89

We observe that:

In ticket 1, there is an offset of -1 of the winning combination
In ticket 2, we have the reversed digit of the winning combination
In ticket 3, we have half of every number in the winning combination
In ticket 4, (this ticket was chosen randomly by the terminal selling the ticket, but studying the listing of the randomly generated combination by the computer we find that this combination came out right before the winning combination (say if computed 1 millisecond later the computer would have printed the winning combination)
In ticket 5, Number 12 is the only number in the winning combination

Question is: With which ticket (1-2-3-4-5) did you came the closest to winning the lottery ?

 

[ThunderChunky]

equal

 

[phildonnia]

Off the top of my head, I think 1-4 are all equally "close", but 5 seems way off. My test for "closeness" would be what are the odds that you would win if, there were a special prize for getting that condition. It seems that 1-4 are all just as unlikely as matching all the numbers.

But lemme think about that for a while.

ETA:

#3 assumes that all the numbers are even. If #3 were phrased as "half of every number, rounded down as necessary", then that condition would be more likely. Now I see that #1 is very very slightly off too, since you can't subtract 1 from the smallest number.

 

[Denver]

The question can make sense, but as implied, it just depends on what measurement or rule you use to define 'closeness'. So perhaps the question is, what rule best defines closeness, and why. And that in turn may depend on what are the rewards for getting that 'close', or how you can adapt your ticket-number generation to get 'closer' by that rule.

In the latter case, since there does not seem to be really any behavior you could modify to get closer by any rule, no answer is better than any other.

But just as a math exercise, I'd choose rule 1.

 

[Jack by the hedge]

I suppose you can define "close" any way you like, but since the game is to match all the numbers, it seems fair to me to say that matching one number is "closer" than matching none. So I might concede that (5) is nearer.

If (4) only missed being the winner by a tiny fraction of a second then yes, that was very close indeed, but a miss is as good as a mile if the required task is to match the winning numbers.

(1) to (3) show patterns which relate to the winner in some way, which make us feel they can't be random, but the relationship is imaginary. The numbers on lottery balls are just identifying symbols; they could equally well be letters or abstract shapes. So any arithmetical association we may make between one symbol and another is irrelevant.

 

[Zubon]

In this case, "close" is subjective. You would have to be more specific on what you call "close". In a lottery, numbers that are not exactly the ones drawn don't count. However, ones with fewer but some correct numbers may get a smaller prize. (e.g. $100 if 3 numbers are correct)

In this case, ticket 5 would be the one that "wins" the most.

Until you redefine "close", this is unanswerable. But interesting on a boring Monday morning.

 

[nimzov]

I voted 4, because this ticket is the closest (in time) to the winning ticket

But maybe now I would chose 5, because in a lottery the only reward is for a "hit" not getting close to a number. Sometimes you even get a small reward for getting 2 numbers right.

 

[theprestige]

Assuming that the only rule of the lottery in question is that one number wins, and all other numbers do not, all five tickets are equally and infinitely far away from the winning number.

As the old saying goes, "close only counts in horseshoes and hand grenades".

 

[nimzov]

(1) to (3) show patterns which relate to the winner in some way, which make us feel they can't be random, but the relationship is imaginary. The numbers on lottery balls are just identifying symbols; they could equally well be letters or abstract shapes. So any arithmetical association we may make between one symbol and another is irrelevant.

Agree on (3) but (1) would be the same as choosing the previous letter or shape, when these letters or shapes are listed in some arbitrary order.

But very interesting point. Thanks.

 

[Bob Blaylock]

I say the whole question is meaningless. There is no “close”. Either a ticket matches all the numbers, and thus wins, or else it doesn't. If it doesn't win, then it doesn't win, period; and there is no point in trying to define by how much it failed to win.

If you got a ticket that was 12 - 14 - 30 - 46 - 52 - 67, it would have failed to win exactly the same as all the other tickets.

 

[blobru]

I voted ticket 5. It's the only one with even one number right. The first three are meaningless patterns in terms of winning (though psychologically, it looks like each is just off target, as if the ticket were a bow and arrows). Four is interesting: temporally, you missed by a millisecond -- maybe the width of the callous on the ticket seller's index finger -- so a good case could be made for it (hmm, now that I put it that way, missed by the width of her callouses, I want to change my vote).

As with any question of definition, it really depends on context: subjectively, 1 - 3; temporally, 4; regulatively (in terms of legal winning numbers), 5.

 

[Cavemonster]

I agree that various definitions of "close" could apply, but #4 and #5 seem to be the strongest candidates.

The number value of the six slots is incidental, it could just as easily choose between the titles of Oscar winning films, or elements in the periodic table. Although we have a mental relationship based on what we know about numbers, that relationship does not apply to these lottery picks which are only incidentally numbers and don't possess any of their mathematical properties.

 

[nimzov]

I say the whole question is meaningless. There is no “close”. Either a ticket matches all the numbers, and thus wins, or else it doesn't. If it doesn't win, then it doesn't win, period; and there is no point in trying to define by how much it failed to win..

If I miss the jackpot by one number (or shape) my guess is that I came closer to winning that if I miss that jackpot by 6 numbers (or shapes).

In my opinion, all loser are not equal.

Let say, I take 100 persons that missed only the last number (5/6), there is necessarily one jackpot winner in within a group of 100 persons (*). But if I take 100 persons (**) that missed the jackpot by 6 numbers (0/6), I have 100 losers and no winner in that group.

Peoples in the first group came closer to winning than people in the second group.

(*) Supposing no 2 players can have the same combination.
(**) or 1 million for that matter

 

[Soapy Sam]

A miss is as good as a missus.
I'd say "equal".

 

[ConspicuousCarl]

Since there is no partial-match prize specified in this example, none of them are "close".

Our idea of non-matching pairs being "close" is a result of our tendency to treat mere strings as if they were values. 99 and 100 have no physical locations, and have no value unless they are determined by some real value.

That is probably enhanced by the ease of imagining some operation to create the correct combination. It would be so simple to just double or flip the numbers! Yeah, but you know what else would have been easy if we were omniscient? Picking the right combination from the beginning. That would be much easier than picking the wrong numbers and then doing math with them.

The time-proximity loser is probably appealing because it is easy to imagine small violations of the laws of physics as being more possible than larger ones, and yet neither has a good record of occurring. If you are going to imagine a microscopic delay in the computer's random number generator, why not imagine that the central lottery computer made a microscopic error and accidentally changed your 99-99-99-99-99 guess to the correct number?

All of this "coulda happened" is silly because what if the winning combination was "12-14-30-46-52-68", but you picked "12-14-30-46-52-69"? Was that "close"? Really? What if I told you that the lottery commission had removed the 69 ball from the machine 20 years ago under pressure from some nutty religious group, and there was no way you could have won short of that ball magically materializing itself inside the machine?

 

[Bob Blaylock]

If I miss the jackpot by one number (or shape) my guess is that I came closer to winning that if I miss that jackpot by 6 numbers (or shapes).

In my opinion, all loser are not equal.

How so? If you lost, because only one number was off by one, then you lost, just the same as if none of the numbers were anywhere close. There's no difference in the outcome.

 

[nimzov]

How so? If you lost, because only one number was off by one, then you lost, just the same as if none of the numbers were anywhere close. There's no difference in the outcome.

If you lose a game of hockey by 6-0 it is not the same as losing by 1-0. Even if the two games are lost games, if you are on the losing side, 6-0 is "farther away from a win" than 1-0. In that sense all loser are not equal.

Now for the lottery. Would you prefer to be part of a group of 100 peoples who had 5 numbers right out of 6 in the lottery I described earlier, or part of that other group of 100 peoples who had 0 our of 6. Which group stand the more chance of having a winner within its members ?

 

[nimzov]

The time-proximity loser is probably appealing because it is easy to imagine small violations of the laws of physics as being more possible than larger ones, and yet neither has a good record of occurring. If you are going to imagine a microscopic delay in the computer's random number generator, why not imagine that the central lottery computer made a microscopic error and accidentally changed your 99-99-99-99-99 guess to the correct number?

No violation of physical laws are involved in ticket 4 argument.

Just to clarify. The "time-proximity argument" is this. Your print out the chronological list of random combination generated by the computer for that lottery, and you find out that the line following your combination (numbers on ticket 4) there is the winning combination printed out. You realized that you missed that combination by a fraction of a second.

 

[theprestige]

No violation of physical laws are involved in ticket 4 argument.

Just to clarify. The "time-proximity argument" is this. Your print out the chronological list of random combination generated by the computer for that lottery, and you find out that the line following your combination (numbers on ticket 4) there is the winning combination printed out. You realized that you missed that combination by a fraction of a second.

But according to the rules of the lottery, missing the winning combination by a fraction of a second is just the same as missing it by a day and a half. In either case, your bank account is no closer to being full of lottery winnings.

 

[bruto]

I think you need to decide what winning and losing are and what they mean first. A close loss in a hockey game is different from a rout partly because it affects the team differently in ways not directly related to whether the game is won or lost. In addition, points in such a game are incremental. If you get more points in the first half, it increases your chance of winning in the second, etc. To get more points, even without a victory, requires more achievement, and to lose by more points demonstrates greater weakness. A close game can be turned around in the last seconds, where a slaughter cannot. Yet from the standpoint of getting to playoffs or winning a cup, a close loss and a rout are equal, so how you look at a score depends a good deal on context.

But in a lottery there's no extra achievement or greater luck in getting closer to the right number. All wrong numbers have the same chance of coming up and the same consequences if they do. A miss is a miss and a loss is a loss.

The one exception I might make is for the last of the sequences. If the numbers are chosen one at a time, and you lose the first number, then you lose the grand prize no matter how interesting or shapely the losing number is. Your chance of winning the grand prize is reduced instantly to zero and your chance of any prize at all is diminished. But if you get a match on the first number, your chance of winning the grand prize is at least momentarily greater than zero and your chance of some other prize at least briefly undiminished, so I would say you come closer to winning if you get a match.

My argument falls apart, of course, if the match is not at the beginning, or if all numbers are chosen simultaneously, whereupon I'd have to say again that all losses are equivalent.