I have just had a disagreement with a friend who thinks that your chance of winning the lottery is greater if your buy five tickets in one week than if you were to buy one ticket per week over five consecutive weeks.

What does every think about this?

He's correct, but the difference will be insignificant when the chance of winning is small. When you buy 5 different tickets in 1 week, you have 5 times the probability of winning the grand prize than if you bought 1 ticket. When you buy 1 ticket per week for 5 weeks, your chances of winning will be very slightly less than 5 times the probability for 1 ticket because of the very small chance that you will win more than once. The 5 tickets in the same week represent mutually exclusive outcomes because only 1 can win, while the 5 tickets over 5 weeks represent independent outcomes. Note that the probabilities would be exactly the same if all 5 tickets were chosen randomly since in that case you could possibly get duplicate winning tickets when you play 5 in the same week, and that has the same probability as winning multiple times over 5 different weeks because then the tickets would be independent in both cases.

I had an example here earlier for a daily lottery to show how this can make a significant difference. That example had a problem in that no matter how you chose the numbers, you could always win on more than 1 ticket on the same drawing, so it wasn't correct to just multiply the probability by 5. Here is an example that should work.

In the daily Pick 3 lotto, you choose 3 numbers from 0-9 in a particular order, and 3 numbers are drawn in order where numbers can be repeated. If you pick 3 different numbers, you can win by matching just the first 2, just the second 2, or all 3 numbers in any order. The probability that you win a prize on a single ticket when you play 3 different numbers is 2.4% since there are 9 ways to match the first 2 only, 9 ways to match the second 2 only, and 3! = 6 ways to match all 3 numbers, for a total of 9+9+6=24 ways to win out of 1000. If we buy 5 tickets on the same day and choose the numbers so that no ticket has the same first 2 or last 2 numbers, like (012, 234, 456, 678, 890), then our probability of winning a prize would be

5 * 2.4% = 12%.

But if we buy 1 ticket per day for 5 days, our probability of winning a prize will be 1 minus the probability of losing on every ticket, which is

1 - (1 - 0.024)⁵ =~ 11.4%.

So our winning chances have dropped. Our EV will still be the same because in the second case we can win on more than 1 ticket.

Of course you usually only win about $25 when you win which isn't what most people mean by "winning the lottery". The probability of winning a large prize in a lottery will be much smaller, and the difference in the probabilities will become miniscule because the chances of winning on more than 1 ticket will be very small.

Intuitively, if there is a lottery that has one million numbers at $1 each you can purchase all numbers for $1,000,000 and be a guaranteed winner -- but there is no way you can distribute $200,000 a week for five weeks and have a guaranteed win.

Right. Happy Mutually Exclusive Day.

Chance is 67%, a bit more than I guessed first. And obviously the aggregate numbers used don't have to cover every number, it doesn't matter.

I'm wondering, in your scenario does the chance to split change at all? I think not.

If the chance is 1/n and you do it n times, the probability has to be greater than 1 - 1/e which is about 63.2%. It converges to that pretty quickly, but it's higher for small n. For n=2 it's 75%.

And to you and yours