Wednesday, April 20, 2005
Lottery tickets and mathematical puzzlers.
Since I'm not sure who I want to slap upside the head next, I'll leave that alone for the time being and toss out another math teaser. Most folks are familiar with the "6-49" style of lotteries where, in this case, you buy a ticket for the chance to pick six different numbers of your choice out of the possibilities of one through 49.
Come draw time, six numbers are selected (allegedly randomly) and, if you have all six winning numbers, you get a fistful of cash. If it turns out that more than one person selected exactly the same six winning numbers, all winners share the pot equally. (Some variations have smaller prizes if you get at least some of the numbers right, but that just complicates life so let's ignore it for the time being.)
Naturally, lots of folks have "systems" that they think increase their chances of winning, and lots of other folks just pick numbers that have some kind of personal significance or sentimental value. But, as we all know, in the end, if the winning numbers are really drawn randomly, then any set of six values is no better and no worse than any other set of six values.
Or is it?
In fact, if you're clever, you can get a sneaky advantage over others playing the same game. Based on just the rules you've read above, how can you maximize your expected return from playing this game? No tricks, it's all above board. Think about it. And if you're stumped, go back and read the challenge carefully. The first correct answer will, of course, win absolutely nothing. I don't get paid enough to give out actual prizes.
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Don't buy a ticket, and your expected return is zero. Much better than the expected return when buying tickets
All right, that's devilishly clever but, no, in this case, you do have to play. Not playing is not an option.
Man, you folks can be such pedants sometimes. :-)
On the assumption that the numbers are actually chosen randomly, pick six consecutive numbers, like 1,2,3,4,5,6. No one will pick obviously "patterned" numbers because they don't look "random", though the odds are just as good they will come up. That way, if you win, you won't split the jackpot.
I have definitely got to come up with harder puzzles.
No one will pick obviously "patterned" numbers because they don't look "random"
But you just did!
I don't buy it. Shouldn't you also take into account that some similarly-minded wise guy out there has also thought along those lines and picked the same numbers? As J. Jonah Jameson would say, "What are the odds?"
I could propose a strategy such as picking a random set of numbers all greater than 31 so as to not conflict with people whose lucky numbers include their birthdays. But, again, what are the odds that someone else out there has the same idea, thus increasing the odds that we'd conflict?
In fact, any attempt to use some sort of system probably has a lower probability of success, I would imagine. In which case, your best bet is to go with a purely random set of numbers.
There is no "system" to increase your chance of winning the grand prize in a randomly drawn lottery. Choosing number sets "not as likely" to be chosen by others may increase you chance of not having to share the grand prize with others with the same numbers. But, as has been discussed above, whatever clever scheme you employ will likely have already been put to use by others who don't understand either.
Read the question again carefully. I wasn't asking how to increase your chance of winning the prize -- that is, picking the correct numbers. As we already know, there's no way to change that, assuming the numbers are drawn randomly.
I was asking how to "maximize your expected return", and the only way to do that is to try to pick a set of numbers that others aren't as likely to choose, which means it's less likely you'll have to share.
Human nature being what it is, very few people would ever consider picking the numbers 1,2,3,4,5,6. "Oh, come on," they'll say, "what are the chances of exactly those numbers being picked?" Then they'll happily pick a different set of six numbers, not appreciating that the chance of those six numbers coming up are exactly the same.
Like I said, it's just a human nature thing.
Are you saying that out of the 20-30 million tickets sold each week that you are the only smarty pants who can see that 1 2 3 4 5 6 is just as likely to be drawn as any other combination?
I think you may want to read up on human nature a bit more.
Gosh, did I actually say that? Let me check ... why, no, I said no such thing. Fancy that.
The point is (and pay attention so I don't have to explain this twice, please) that many people will never choose a set of numbers like 1,2,3,4,5,6 because, in their minds, it's just too improbable. They see the pattern and think, oh, that can never happen. And they fail to appreciate that it's no less likely than any other set of numbers.
I never claimed that I was the only one clever enough to know that. That's not necessary. All that's necessary is for a sufficient number of other people to think the other way.
Remember, these are the same people who, after watching a flipped coin come up heads five times in a row, are really, really sure that it's time for a tails. Is that clear enough?
And, yes, it really is about human nature.
Your chances of winning the lottery are only -slightly- increased by actually buying a ticket! ;)
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