# Probability question wrt to lotteries

Lottery players often buy multiple lines or tickets to improve their chances. Typically from what I’ve seen, players will buy different rows of numbers rather than rows that are all the same, thinking this will improve their chances. However, wouldn’t they improve their chances equally well by using the same numbers for every chance? I wonder about this because there was movie years ago in which a couple won a lottery, with the total jackpot of, like, \$30 million, but found out next day that they would only get about \$3 million, because “some bowling team” had won the rest. This means that the bowling team would have had to play the same numbers for each line that they bought, instead of differing ones which is what groups usually do.

This theory seems to work in a hypothetical small case. If you have 10 people playing, and nine of them cooperate and all pick the same numbers, then there’s a damn good chance that the numbers drawn will match one of those tickets. But what about a real life scenario? In a state lottery draw, does the bowling team above have the same chance of winning if it selects the same number over and over for nine lines, as opposed to randomly picking differing sets of numbers?

If I understand the scenario correctly, if you and I both play the lottery, and I pick the same set of numbers as you, I have the same chance of winning as if I had picked a different set of numbers. But if I do win, I have to split the pot with you. So there is no advantage to picking the same numbers as anyone else.

My guess is that the writers of the movie in question didn’t think about what they were implying all that carefully, assuming our interpretation of the situation is correct.

And if my chances of winning with any particular set of numbers is, say, 1 in a gazillion, then if I buy 7 tickets and play sets of numbers with each, I have a 7 in a gazillion chance of winning the jackpot (which I would have to split with anyone else who happened to choose that same set of numbers). If I bout 7 tickets and played the same set of numbers on each, my chance of winning the jackpot would only be 1 in a gazillion, but if I did end up winning but having to share the jackpot, I’d get 7 shares, one for each of my tickets.

Picking the same numbers on multiple tickets doesn’t help your odds of wining. You only have one chance to win – if all your numbers come in (I’m skipping partial match winners).

However, if you do win, you can win more – *if *there are other winners (e.g., assuming you buy nine tickets all with the same numbers, you will get 9/10s of the money if there’s a second winner; if you buy a single ticket, you get half the money).

Picking different sets on numbers means you have more possibilities. You if you have nine different combinations, you have nine times the chances of winning than if you have just one combination. However, if you win, you will have to split the winnings 50-50 if there’s a second winner.

If you’re the only winner, it shouldn’t make any difference in the payoff; the nine winning tickets would each give you 1/9th the payoff.

It’s also easy to show that picking different numbers provides a much higher expected value, and that picking the same number has diminishing returns.

If we assume the chance of winning on any given set of numbers is p, the number of tickets you’ll buy is n, and the number of other people expected to have picked the winning set is m, and the cash prize is C, then the expected value of picking n different tickets is

Cpn / (m + 1).

The expected value of picking the same set on n tickets is Cpn / (m + n).

Another thing to keep in mind; because so many people have mistaken ideas about randomness (e.g. that 11 12 13 14 15 16 is too “non-random” to possibly come up if the drawing is fair), you should choose sets of numbers that have an obvious pattern. It won’t improve your chances of winning, but it will improve your chances that if you win you won’t have to share the winnings.

By the same token, Powerball numbers go up into the forties. If a large percentage of people use birthdays as a basis for thei numbers, picking numbers from 32 up will cut down on the chances of sharing.

This has sort of happened twice that I know of in Australia. On one occasion a woman forgot that she had entered lotto and entered again using the same game numbers. She won and, being the only winner, won half the jackpot twice. On the other occasion a mother and daughter had each used the same family birthdays on a family entry that they had used for years and they came up. There was another winner so they got 2/3 of the prize pool and he got 1/3. They had never noticed that two games on the coupon used the exact same numbers.

On the other hand, many people DO have favorite lottery numbers that DO form a pattern. Lots of people buy 50, 49, 48, 47, 46, or patterns that look geometric if you lay the available numbers out in a grid.

If you are trying to develop a strategy for improving your share of the payoff, you don’t need to worry about the people who pick, or let the machine pick, essentially random groups.

The people you’re trying to out-guess are the ones who pick their own specific numbers for some (unknown to you) specific reason.

Among that population, I’d bet more folks pick numbers which have a discernable pattern than pick numbers which are random to their best ability.

Why do I say that?

I do know from years of watching people play Keno in Las Vegas that there is a very strong bias for them to pick patterns (and birthdates).

I also know there was a lottery win a few years ago where a decent sized (\$1-\$10M) pot was won by the combo 1, 2, 3, 4, 5, 6. Tens of thousands of people shared the prize, so each got a comparative pittance. (Naturally Google is no help finding a cite, but boy can they help with finding places that’ll sell you a “guaranteed” number picking system.) So there’s one (soft) data point that lots of people do pick patterns.
Since you want to pick whatever most people don’t, you therefore want to pick randomly, rather than in a patttern.

Bottom line: it all hinges on which population you think is smaller: random pickers or pattern pickers. As I say, my opinion is the smaller population is the random pickers so that’s what I’d do to maximize my theoretical winnings. I admit I have no citable hard data on actual lottery ticket purchases to support that opinion.

There was a famous case in Massachusetts a number of years back, involving a game called MassCash, which paid out \$100,000 to everyone who had the winning number.

One time the winning combo was the set of numbers you would get if you used the numbers at the top of the five columns on the betting sheet to make a “V” shape. There were so many winners, the prize pool wasn’t big enough.

This page gives the details. I don’t see how those numbers made a “V” shape, but maybe I’m remembering that detail wrong.

My favorite lottery strategy for big-spenders came out of a book of collected columns I saw from some guy who wrote about lotteries.

Let’s say it’s six-number game. You pick a field of 8 numbers, and play all 28 six-number combinations of that.

If your numbers are 1,2,3,4,5,6,7,8, you play:

1,2,3,4,5,6
1,2,3,4,5,7
1,2,3,4,5,8
1,2,3,4,6,7
1,2,3,4,6,8
1,2,3,4,7,8
1,2,3,5,6,7
1,2,3,5,6,8
1,2,3,5,7,8
1,2,3,6,7,8
1,2,4,5,6,7
1,2,4,5,6,8
1,2,4,5,7,8
1,2,4,6,7,8
1,2,5,6,7,8
1,3,4,5,6,7
1,3,4,5,6,8
1,3,4,5,7,8
1,3,4,6,7,8
1,3,5,6,7,8
1,4,5,6,7,8
2,3,4,5,6,7
2,3,4,5,6,8
2,3,4,5,7,8
2,3,4,6,7,8
2,3,5,6,7,8
2,4,5,6,7,8
and
3,4,5,6,7,8

Your odds of winning the jackpot aren’t all that much greater, BUT:
Every 3-number combination appears 10 times
Every 4-number combination appears 6 times
and
Every 5-number combination appears 3 times

So if you score one of the lesser prizes, which is slightly more likely than winning the jackpot, you win it multiple times.

Let’s say the winning number is 1,2,3,4,9,10

You win the 4-prize with your entries
1,2,3,4,5,6
1,2,3,4,5,7
1,2,3,4,5,8
1,2,3,4,6,7
1,2,3,4,6,8
and
1,2,3,4,7,8

And the 3-prize with your entries
1,2,3,5,6,7
1,2,3,5,6,8
1,2,3,5,7,8
1,2,3,6,7,8
1,2,4,5,6,7
1,2,4,5,6,8
1,2,4,5,7,8
1,2,4,6,7,8
1,3,4,5,6,7
1,3,4,5,6,8
1,3,4,5,7,8
1,3,4,6,7,8
2,3,4,5,6,7
2,3,4,5,6,8
2,3,4,5,7,8
and
2,3,4,6,7,8
1,3,4,6,7,8

Of course, you have to drop \$28, and you may easily still walk away with nothing, but it’s still sorta cool.

Right, I didn’t think you would have a better chance by playing the same numbers on multiple tickets, rather, the same chance as if you bought the same number of tickets playing different numbers. My question boils down to, “are the chances equal in both scenarios?”

No.

Example Lottery with numbers 1-4. You choose 2 to win.

If you choose

1 2
1 2
1 2

Then you have a single chance of hitting. i.e. if the numbers drawn are 1 and 2.

If you choose different numbers

1 2
2 3
3 4

Then you have 3 chances to hit. Your chances of winning have tripled because you played three different combinations.

One more data point: In the Colorado state lottery (5 million to one odds), the numbers drawn once made a pattern, four corners and the two in the middle (on the fill-in-the-numbers-on-the-grid sheet) and 11 people hit the jackpot. I don’t remember any other time there were multiple winners (altho’ I’m sure it’s happened).