robin wrote:In my (no doubt wrong) view of it, the two events are mutually exclusive - by definition because you've said that the tickets are different. The probability of both winning is zero - thus they are mutually exclusive.
Correct assertion...
robin wrote:
Hence for the lottery with two different tickets you get 1/big-number + 1/big-number = 2/big-number.
...wrong conclusion. (IMO, I think, perhaps...)
So, sticking with the more narrative "odds" notation, and just looking at jackpot odds:
You pop down to the corner shop and buy a ticket at odds of 1 in 13,983,816. You're about to leave, but decide to go back and buy a second ticket with different numbers to double your chance of winning. You go home happy, you place the two tickets in front of you on the table and stare at them. Each ticket has an equal chance of winning, and each ticket has odds of 1 in 13,983,816.
Two tickets twice as "good" as one, right? But each ticket has an equal 1 in 13,983,816 chance. But you've got two tickets, so we're no longer interested in the odds of an individual ticket winning. Lets change the question to what we really want to know... What's the chance of YOU winning? Or, what's the chance of EITHER ticket winning? Ahh, now we're asking the right question. (Most of you will have got this far already... if you win, you don't care which ticket it was).
Analogy time...
I've lost my wedding ring (the jackpot) somewhere on a football field (the number space of all possible lottery ticket combinations). I (Mr Ticket) have to search for it (pick the winning numbers).
If I have a friend to help me (another ticket) we'll probably find it in half the time, right? Correct, we will, but how?
By communicating. By saying "You search that half, and I'll search this half." and understanding what that means and sticking to our own half. How long would it take if we couldn't talk or communicate? Well, the only thing either of us would know for sure is that the ring is NOT where the other person is standing, but could be anywhere else. We're both back to searching the full football field, minus the bit that has a person on it. Add another person to help (3 tickets) and how much field do we have to cover? Still almost all of it. Add a dozen people. Have we significantly divided the labour if we can't communicate? Nope, we've barely made a dent on the problem.
This is the problem lottery tickets have. They are totally independent as Robin asserted, and can't communicate. The have no effect on each other. No way of dividing the work. They can't decide to cover half the possibilities each, so each operates independently in the entire space. Except that's not quite true... they did communicate once, through you just once when you decided not buy two tickets with the same numbers. (The same as not looking for the ring in any spot that has a friend standing on it). So, they each know they're not the same as the other, but there's no communication mechanism between tickets that says, "I'll cover this 6,991,908, you cover that 6,991,908". So the odds are NOT 1 in 6,991,908, or 2 in 13,983,816.
So, what are they?
Well, that second ticket isn't the same as the first because when we picked it, it was chosen from a slightly smaller pool of possibilities (13,983,816 minus the first ticket combination, equals 13,983,815 remaining possibilities). We can shuffle the tickets around and it doesn't make any difference which is which or which order we bought them in, so while the odds of a particular ticket winning remain at 1 in 13,983,816, the odds of YOU (either ticket) winning have dropped to the odds of your "second" ticket, or 1 in 13,983,815. Buy another (different) ticket and it will drop to 1 in 13,983,814.
The odds of any ticket (i.e. you) winning the jackpot are simply 1 in (total possibilities - number of tickets bought). There's no division of labour. No dividing up the search space. Every ticket is scrabbling around looking for the jackpot in the entire space of possibilities, except where there's already another of your tickets standing.
So, in the case of two tickets, for that extra £1 (effectively doubling your stake), you increased your chances by a whopping... 0.0000071524%.
