What are the odds of winning a lottery, if I insist on always playing a certain number?

Question

It has been a while since I was out of school so I am hazy on stuff like factorials or binomial coefficients. So I need an explanation on how to use them for my problem.

I want to join a lottery where I have to pick $4$ numbers out of the numbers $1$ to $39$. The order of these picked numbers does not matter. To keep it easy on myself to check results, I always use the number $16$ as one number. I know this reduces my odds of winning the lottery, but I don't know how much this reduces it as opposed to picking all $4$ numbers at random. (I guess we can assume that all $39$ numbers have the same odds of getting picked.)

Any ideas would be much appreciated.

Answer 1

There are ${39 \choose 4}=82251$ possible draws. The lottery tries hard to make them all equally probable. To the extent they succeed, any set of numbers you pick has the same chance of winning, so always picking $16$ as one of your numbers does not change the odds.

Even if you buy two tickets having $16$ on both of them does not change your odds of winning a specific prize that depends on matching all four numbers. It can make it less likely to win some prize in a lottery that offers more than one by making it possible to win more than once in a given lottery.

The only thing you can do to improve your situation is to pick an unpopular set of numbers to reduce the number of ways you split the prize should you win. There is a variety of advice on how to do that, but I don't know how much of it is based on study of what people do and how much of it is speculation.

Good luck with $16$

Answer 2

All possible combinations have the same probability, but as Ross Millikan explains in his answer, you can consider whether other people are likely to choose the same numbers. As pointed out here, the human brain is a poor random generator, it's easily influenced and that in a far more deterministic way than we would think is possible.

Your preference for the number 16 and the other numbers you are going to choose, could thus well be the result of being exposed to information that many other lottery participants are also exposed to. So, other people could well have chosen the same numbers. Therefore it's best to choose lottery numbers by drawing them randomly from a set that excludes your favorite numbers and number combinations.