Does playing 100 tickets once monthly give a slightly better chance of winning, than buying one ticket daily for 30 days?

Question

Do you stand more, less, or the same chance of winning a lottery jackpot by playing 100 tickets for one draw once a month? Or by buying one ticket every day for 30 days?

My initial assumption is that by playing 100 tickets once a month, you have moved the decimal point over two positions and thus stand a better chance with 100 tickets once a month. However, with one ticket every day for 30 days, you face the same astronomical odds each day. Is my assumption correct that playing 100 tickets once a month gives a slightly better chance of winning than buying one ticket every day for 30 days (the same game throughout, of course, in both instances)?

Answer 1

Well, in one case, you're buying $100$ tickets for a single lottery, once a month, while in the other case, you're buying about $30$ tickets a month, so your chances are more than three times as good in the former case.

But imagine that you compare buying $30$ tickets for a single lottery, once a month, against buying one ticket a day, each for a single lottery, for the thirty days in a month. Then it's more of an apples-to-apples comparison.

Suppose that the lottery is a lotto: Each ticket has a one-in-$N$ chance of winning. We'll also assume that you're buying $30$ distinct tickets in the former case. Then the odds of being a winner in the former case is $\frac{30}{N}$, while in the latter case, the odds of being a winner are $1-\left(1-\frac1N\right)^{30}$, which is ever so slightly less than $\frac{30}{N}$. The difference is made up for by the fact that you have an even more microscopic chance of being a multiple winner, which you obviously can't do in the former case.

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On the other hand, suppose that each day, there's exactly one winner, and $M$ other players. So in the former case, on the day that you play, you have a $\frac{30}{M+30}$ chance of winning.

In the latter case, you have a $\frac{1}{M+1}$ chance of winning each day you play, but you play $30$ times, so your overall chances of winning are $1-\left(1-\frac{1}{M+1}\right)^{30}$, which is this time slightly higher than in the first case (for large values of $M$)—with the difference being accounted for by the fact that you're not buying a bunch of tickets with diminishing returns.

Answer 2

yes -- odds are the same per ticket, no matter when you buy them (provided it's the same lottery). So buying 1 ticket per draw over the course of 100 draws gives (technically) the same odds of winning as buying 100 tickets of the same draw HOWEVER, if you skillfully selected different numbers in all of those 100 tickets that you bought to play on just one draw, then that would give you better odds than if you were to buy 1 ticket in each of 100 different draws. IN both cases, buying only 30 tickets gives lower odds of winning.