# Settle a bet about lottery odds.

The disagreement is:

One person says you’ll have better odds buying 10 tickets for 1 drawing rather than buying 1 ticket for 10 drawings.

The other person says it’s the same exact odds either way.

Who’s correct?

It depends on what you’re evaluating. If you’re just evaluating the probability of winning so much as a single lottery, then the first person can have better odds (imagine that each drawing had only 10 tickets total, for example, out of which 1 would win). Specifically, if you’re looking at drawings with K tickets total, then the first person has a 10/K chance of winning at least one, while the second person has a 1 - (K-1/K)^10 chance of winning at least one.

However, if you’re looking at the expected gain (i.e., the arithmetic probabilistic mean winnings), then they come out equal: the fact that the second person is less likely to win at all is counterbalanced by the fact that he can win more than one drawing. If each drawing has K tickets total with a payout of P, then both will earn an average payout of 10/K * P.

Depends on how you look at it. For instance, the simplest lottery I’m aware of is something like the Daily Number, 3 numbers pulled.

The odds that you win with any single ticket is 1 in 1000. That’s easy enough. However, if you have 10 tickets, you have a 10 in 1000 chance that your number will come up.

But…as each number is independent of the others, you’re looking at 10 individual 1 in 1000 chances of winning. A wrong ticket doesn’t increase your chances of winning on the other 9.

EDIT: Well, I know what I’m trying to say, but in retrospect I don’t know whether I said it right. I’ll let the mathematicians sort it out.

10 tickets for 1 drawing are 10 in a gazillion.
1 ticket for 10 drawings are 1 in a gazillion for each drawing.

Let’s simplify. Suppose the lottery has just ten tickets @ \$1 each, and the prize is \$5.

If you buy ten tickets in 1 drawing, you will win one prize and get exactly \$5 back.

If you buy 1 ticket in each of 10 drawings, then:
(1) you have a 34.867844% chance of getting no prize
(2) you have a 38.7420489% chance of getting one prize
(3) you have a 19.3710245% chance of getting two prizes
(4) you have a 5.7395628% chance of getting three prizes
(5) you have a 1.1160261% chance of getting four prizes
(6) you have a 0.1634937% chance of getting five or more prizes
But you have an expected value of \$5, i.e., if you did this many times, on an average you would win 1 prize of \$5.

Everybody seems to be answering this for a lottery which consists of sold tickets exactly one of which will be drawn. But the most common lotteries now – the state and multi-state lotteries don’t have tickets like that. They just pick numbers at random with no guaranteed winner.

With the state lotteries the difference comes from multiple wins. If you pick in ten different lotteries, then you could win the grand prize or any particular prize zero through ten times.

If you pick ten times in one lottery and if you pick different sets of numbers then you could only win the grand prize once (so correspondingly you’d have a slightly larger chance of winning at least once). If you picked your sets of numbers completely at random (allowing for the possibility of identical picks), you’d make your chances of zero through ten prizes identical to tien picks in different lotteries. (However, under teh rules of most (all?) of the lotteries, you’d share the grand prize with yourself not double up on it so your exepcted winnings would drop.)

For the fixed-amount lesser prices. (E/eg/. match 4 numbers win \$1000), picing at random in one lottery would have an identical pattern of wins to picking in ten lotteries. If you picked non overlapping sets of numbers, your chances of winning at lesat one fized prize would go up, and your chances of winnning multiple fixed prizes would drop (to zero unless you can win a fixed prize with less than half of the numbers), but your expecte winnings would remain the same.

It actually doesn’t change the analysis too much. Let’s say that there’s a lottery with a prize of \$5 where ten tickets are sold to the general public and the lottery agency retains ten more. If you buy all ten tickets, your expected prize is \$2.50. If you buy one ticket in each, your expected prize is still \$2.50.

It seems to me that OldGuy is on the right track. “The lottery” these days usually means a big government run thing without a guaranteed jackpot winner in any given draw. The odds of getting all the numbers right are the same regardless of how many other tix are sold. Hold on, that’s not my answer, that’s just the rules.

For example, in the Hoosier Lotto, you pick, or let the computer pick, 6 numbers from a field of 48.

The odds for the Jackpot are 1:12,271,512 (picking all 6 correctly)
The odds for a free ticket are 1:7.31 (the smallest prize, for picking 2 correctly)

In this set of facts, the OP seems to ask us to compare buying 10 tix in tonight’s draw with buying one ticket in 10 draws. Now, there are three possibilities for answers:

One school of thought says every ticket has the same odds, whether you buy one or 10,000 tix.

Another group insists that 10 tix in one draw gives you 10: 12,271,512, which is microscopically better than 1:12,271,512.

The Fran Liebowitz school of thought says your odds of winning the lottery are the same whether you buy a ticket or not.

As a mathmetician, I am a pretty good mower of lawns. I’m only posting this to narrow down the field of play; I don’t think the OP was asking about raffles and small scale lotteries.

If this lottery follows typical lottery rules, then in the first case, you buy ten tickets in the same drawing, each with a 1 in 10 chance to win \$5. However, if more than 1 ticket “hits”, they share the \$5 prize. So, by your second calculation, you have a 34.8678% chance to not “hit” anything. That gives a 65.1322% chance to win \$5. Therefore, your expected winnings are only \$3.26. You’d be better off with 10 separate drawings in that case.

While that does seem to be the typical way that the the big lotteries are run in the United States, I’m not sure that it’s a universal rule for lotteries. In addition, the rule about sharing prizes bth makes calculations harder, and reduces the expected value of a lottery ticket – one more reason not to buy a ticket.

Your chances of winning the lottery are only marginally improved by actually purchasing a ticket.

I look at lottery tickets as a cheap way to fantasize about winning, with no expectation of actually doing so.

I don’t know where I heard it but " The lottery is a tax on people who are bad at math" is how I view it.

I hate that expression. Is insurance a tax on people who are bad at math?

Although I do appreciate two equally irrelevant responses, I was looking for actual answers…not impish opinions. Thanks. :rolleyes:

Yes, if you can afford to buy the insured item outright, it is.