The odds of winning the top prize in the UK Lotto are roughly 14 million to 1 (pick 6 numbers from 49). What I want to know is, if I buy two tickets, i.e., two sets of 6 numbers, what are my chances of winning. It would seem that my chances have doubled but what are now the odds of winning the big one?
Your odds are roughly 7 million to 1.
Odds and probability measure the same thing but in different forms.
Probability = number of ways to win / total number of combinations.
So your probability goes from 1/13983816 to 2/13983816.
Odds = number of ways to lose: number of ways to win
So your odds go from 13983815:1 to 13983814:2, which is the same as 6991907:1 or close enough to 7 million to 1.
Of course there’s a 1/13983816 probability that your two quickpicks are identical, so it will slightly less than halve your odds - by 1/13983816.
Or… 2/13983816 - 1/(13983816)^2 I think
If your quickpicks are identical, your odds of winning don’t change. Why would they? You’d be back at 14 million to 1.
The expected value of your gamble does, however, change, since you’ve put twice as much money on the same ticket but won’t get twice as much money if you win the jackpot (though you may get twice as much of any lesser prizes).
I think that’s the possibility md2000 is including in the calculation.
Unless there are multiple jackpot winners - in this case, if you have two identical winning tickets, you get two slices of the pie.
Yes. you buy two quickpicks. (Sounds like the Canadian Lotto-649)
There’s a 1-14million that the two quick-picks you buy are the same ( or 1/13983816)
and 13983815/13983816 that you have two distinct tickets.
So your odds are 1/13983816 That you’ll win if you hit the 1/13983816 odds of duplicates, or
1/(13983816)^2
and your odds are 2/13983816 to win if you had the 13983815/13983816 chance to buy distinct tickets.
It’s so much easier to write with cut and paste.
Ah, I see. You’re reducing the overall probability of winning by the probability of getting duplicate tickets. Ok, that’s clear to me now.
Wouldn’t your chances of winning with two tickets (not allowing for duplicates) be the reciprocal of your chances of losing? I.E., chance of losing with two tickets is (13983815/13983816)^2. For very low odds the chances of winning with two tickets are close to but not exactly your chance of winning with one ticket X2.
I think I was expecting the answer to be closer to “almost makes no difference” than buying two tickets halves the odds.
From Great Antibob’s answer if I buy 1000 tickets the odds of winning are 14,000 to 1. This would make it almost worthwhile to setup a super syndicate when the jackpot is a high amount. However, there is no denying that if I buy 7 million individual tickets then the chances of winning are 2 to 1.
It’s been tried a few times, and it’s sort of worked. But it’s a lot of work and the returns aren’t that great.
The real problem is that you don’t play the lottery in a vacuum. When the jackpot gets high, more people play, thus increasing the chances of a split jackpot. You still have to be lucky enough to avoid split wins, and relying on luck is what you are trying to avoid.
You’re never going to see a positive return on investment playing the lottery.
Unless you win.
I’m sure Antibob means the mathematical ‘expected return’ of playing the Lottery.
This is always negative, since (for example) the UK Lottery only pays out around 45% of the entry fees (the rest goes on profits, operating expenses and taxes.)
So yes, some people win. But most lose - and the ‘expected return’ on your ‘investment’ is -45%. :eek:
Hmm. But the national lottery isn’t a straightforward “win the jackpot or win nothing” scenario is it. For a single £2 ticket I could win back my stake plus an increasing prize amount by matching 3, 4 , 5, 5 plus a bonus ball, or 6 numbers.
So if I define winning as simply getting a positive return on my investment then (according to the national lottery site) I have about a 1 in 54 chance of “winning”.
OK, I’m sure they’re right. But what does that calculation look like?
Yes, I meant as a long term investment strategy. Even with a large jackpot, it’s an overall losing proposition (in an expected return sense) to organize a conglomerate of people to purchase every possible lottery combination.
In terms of making money from it, sure, you can get lucky and win. I could also be left a massive inheritance by a long lost relative.
Neither is a good long term bet.
Still an overall loser for a player. Give us the numbers, and we can figure out just how bad, but it’s not a positive expectation (i.e. on average winning) game.
Note that this is what I meant about split jackpots. If you don’t have to worry about splitting jackpots, several lotteries turn into winners (in the expected value sense) when the jackpot gets large enough. But once you have to worry about splitting jackpots, you have to model the probability of doing so. And any reasonable probability of splitting makes an on-average-winning expectation unlikely.
I suspect that’s not true. I suspect there are combinations that are very unlikely to be chosen by others so that if you pick them you are unlikely to match anyone else if your numbers happen to be chosen. We do know for example that people tend to pick numbers less than 31 since they like to pick dates. And while some one might pick 1 2 3 4 5 6, I’d guess that something like 33 34 36 37 39 40 would be unlikely to be chosen.
OTOH, I don’t know what fraction of people let the computer pick for them. I assume the computer does pick randomly so those would be just as likely as any other 6 numbers.
I don’t suspect. I run the numbers. The numbers say it is true.
Yes, the computer picks randomly (or close enough as not to matter).
The number of quick pickers vs own number pickers is about even or slightly in favor of quick picks.
That doesn’t really matter. People who pick their own numbers tend to pick between 1-31, meaning there is be a slight advantage to picking larger numbers to avoid split jackpots. But the advantage is only for split tickets. Split tickets don’t actually matter unless you win in the first place. The problem is, people already know about this (there was a mathematics paper detailing this psychological phenomenon a couple decades ago) and many people deliberately pick those numbers.
So at this point, you’re still looking at needing 3 things: (1) jackpots large enough to produce a positive expectation (happens rarely) and (2) the winning numbers chosen from the more limited pool of larger numbers (happens even more rarely in conjunction with large jackpots) and (3) no splitting (also rare at large jackpot levels).
The combination of all 3 means the expectation is negative. That’s not to say somebody won’t get lucky and win a big solo jackpot - that happens all the time, but the value of the average ticket in those big wins is less than the cost of purchasing the ticket.
And when you factor in taxes and the loss due to the lump sum (unless you opt for a multi-year payout), it just gets worse.
So, no, it’s not strictly impossible for the lottery to go positive expectation, but it requires people not to get excited and buy tons of tickets when the jackpot gets big. That just doesn’t happen.
No this isn’t true. We’re talking about whether then conditional expectation of playing rarely picked numbers when the prize is very large has a positive expected value. This expectation does not depend on the probability of (1) or (2) since those are conditioned out.
The only issue is how big is the prize and what’s the probability that your set of numbers is picked by someone else. If, for example, the prize is $30 million for a $1 ticket, there are 20 million different possible picks, and the probability that one other person has picked your exact numbers is 40%, the probability that two others have icked your numbers is 10%, then your expected payoff is (1/20)[(0.5)*30 + (0.4)15 + 0.1(10)] = 1.1.