With the big Mega Millions jackpot, I decided to set fire to a few dollar bills for some tickets.
About half of my number sets were quick-picked, and it ended up that there was quite a bit of overlap (i.e. among six sets of numbers, there are only four different lowest numbers).
My instinct is that this set of numbers is less likely to win than one whose number sets are more spread out, because I have fewer chances than that other hypothetical set should the lowest number picked not be one of those four (ie “If the lowest number picked isn’t 2, 6, 8, or 18, I’m boned, whereas the guy whose lowest numbers are 1, 19, 20, 21, and 43 still has a shot — or has a better shot.”).
On the other hand, I know that each set of numbers is equally likely to win.
Please explain in layman’s terms why my instincts are wrong (or how I’m somehow right, if it comes to that).
If you’re just talking about winning the big prize where every number must match, you’re wrong. Suppose I just told you the lowest number picked. You’re correct that anyone with 6 different lowest numbers would be less likely to have no chance than you would have since you only have 4 different lowest numbers.
On the other hand, suppose the lowest number is 19. The other person with one of those tickets still has a very small chance that the remaining numbers match up. But if the lowest number picked is your 2 (and that’s one of the numbers for which you have two tickets) you have twice the chance to win based on just knowing the lowest number.
Your higher odds of being completely disappointed when the lowest number is picked is exactly offset by your higher odds of being in twice as good shape if one of your “doubles” is picked.
I didn’t even think about that. That makes a lot of sense.
And yes, I was talking about the jackpot. If we’re talking about more minor prizes, then my numbers do have a lesser chance, right? If so, how much are we talking about here?
Your original flaw is trying to fit a pattern onto randomness. That’s very human but has no meaning. Random numbers don’t care about patterns or lowest number or consecutive numbers or anything at all.
Casinos make a lot of their money because they know people will think as you do, and will bet in a losing way as a result of trying to make the numbers mean something.
You were right when you said that each set of numbers is equally likely to win. Stop there.
Agreed, but the OP was talking about having a set of tickets where the same number comes up on more than one ticket - and OldGuy already nailed the answer.
Of course, if you pick the exact same set of numbers twice (and I’ve seen somebody pick the same set five times in one drawing), that is kind of a waste. The only way that would pay off when it comes to the jackpot is if somebody besides you also wins – it would give you a larger share.
The money and odds for the lesser prizes are worth considering, by the way. When considering those, the overlapping numbers won’t hurt the average amount of money you win, but they will increase the risk that you’ll win nothing. Spreading out the numbers is similar to diversification in the stock market – the goal is to reduce risk. I mean, you’re very likely to lose money on the lottery ticket “investment,” but still…