Many people know that the odds of winning the lottery are astronomically low with the probability of winning a standard 49 ball lottery with six numbers to pick being 1/13,983,816 or 0.00000715%. But what if that wasn’t the only way to win? As well as the jackpot, prizes are also given out for guessing three or more numbers correctly, with prize values increasing the number of numbers matched, meaning that it is possible to win at least some money without getting half of the numbers required for a Jackpot.

Photo by Alejandro Garay on Unsplash

For any given lottery, we can calculate the probability of getting any particular result using the following formula.

Defining n = number of possible balls, p = number of balls that you pick, and c = number of balls that are chosen correctly, we have:

\frac{1}{c!n!(n+c-2p)!}\left(\frac{p!(n-p)!}{(p-c)!}\right)^{2}

In our lottery example we will be working with n = 49 and p = 6, transforming the equation into one with only a single variable.

Number of correct guessesProbabilityAverage number of plays until winning
043.5965%2
141.3019%2
213.2378%8
31.7650%57
40.0969%1,032
50.0018%54,201
60.000007151%≈14 million

There is a less than 2% chance that you will be lucky enough to win a prize in most lotteries and it will be so small that it is hardly worth playing. However, what happens if the lottery is poorly designed?

The Winfall Lottery ran from 2003 to 2005 in the state of Michigan, USA and featured a very flawed mechanic. If the jackpot reached 5 million without being claimed, it would be distributed to the smaller prize winners, resulting in about a 10 times increase in prize money for all categories. This doesn't seem like a huge deal at first; even with the 10 times increase in prize money, most players would still make a loss, since with a starting bet of1 it would take an  average of 57 dollars to make back50 or 1,032 to make back1,000. However, one couple figured out a cool mathematical trick to increase their odds of winning big.

Number of matched numbersNormal payoutPayout with distribution
35</td><td class="has-text-align-center" data-align="center">50
4100</td><td class="has-text-align-center" data-align="center">1000
52,500</td><td class="has-text-align-center" data-align="center">25,000
6JackpotJackpot

They realised that matching 4 numbers would give them the best return for their investment. By carefully choosing their numbers so that no tickets shared 4 numbers with any of the tickets, for a pool of about 1,000 tickets they could guarantee that they would get one of the correct combinations.

The lottery however, like most similar games, had prize sharing, which decreases the chance of making big wins with the four ball method.  Even with the 100ish tickets, the possibility of having to share winnings with other people could mean losing money over winning big. Luckily there is a technique that can help to reduce the probability of having to share the prize fund. 

Humans are really bad at simulating randomness, so when someone is asked to come up with 6 random numbers, some appear much more often. Numbers found in birthdays, that form patterns on the card, prime numbers and numbers ending in 3 are all much more likely to be chosen. Understanding this and picking the statistically unpopular numbers decreases  dramatically the chance of sharing the prize. The unpopular numbers are in ascending order; 40, 39, 20, 30, 41, 38, 42, 46, 29, 49. All are large, most are composite and all but 3 end in 8, 9 or 0. Keep this in mind for your own lottery exploits.

Read more about the Winfall lottery and the film it inspired.