Are lottery numbers really random?

This question - picked perfectly at random, of course - comes from user syhprum on our forum. Tim Revell explained the maths, as well as how to pick a good set of numbers...

Tim - It's a good question and one that I enjoyed looking into. So that we're all on the same level. The UK lottery is 59 balls numbered one to 59 and you win the jackpot if you correctly guess six balls that are selected from those 59 and your chances of doing that are about one in 45 million. So if you do it, you are a very lucky person indeed.

Chris - How'd you calculate the one in 45 million?

Tim - The number of balls that could be chosen in the first instance is 59 of them. In the second instance there are 58 of them. In the third instance there are 57 and you do that until you get to six balls, multiply them all together.

Chris - So it's one in 59 times one in 58, 57 and so on six times over, and that's where you get that number?

Tim - Yeah. Cause that means you would pick the correct sequence of numbers that comes up. And so what does that, what does it actually mean for the numbers to be random? So some people think that if it was random you would expect that the numbers would come up roughly the same amount of time. And the way the lottery is currently set up has been since 2015, and so since 2015 the number 58 has come up 59 times. And that's the most common number. But the least common number is the number 33 which has come up 30 times. So that's quite a big difference. That's twice as many times that the most common one has come up than the least common one. So does that mean it's fixed? I guess is the question you will want to know the answer to. And the answer to that is no. So even though it is random, you would expect these variations and as the lottery goes on with that same format, you would expect the gap to close at least as a proportion.

Chris - Yeah. Cause the other thing that people do is they choose numbers that are selection or smattering from across the number one to 59 because there seems to be this innate bias in people that the number one to the sequence one, two, three, four, five, six is much less likely to happen than a random smattering. But actually it's equally likely to happen mathematically isn't it?

Tim - Yeah. So on that exact point, just because all of the sequences are likely, that doesn't mean that some sequences are not worse choices than others.

Chris - Because obviously if you choose the same sequence I do, I've got to share my winnings.

Tim - So it's thought that the sequence one, two, three, four, five, six is chosen by about 10,000 people every week. So even if your jackpot is 4 million, that's 400 quid that you get. It's not very good considering on another week you could get all 4 million. So I did, I looked up some tips on the best numbers you should choose. You should avoid picking numbers under 30 because people tend to pick dates so any dates tend to have numbers under 30 in them. You should also avoid lucky number seven because everyone picks lucky number seven so you don't want that one.

Tim  - And people like sequences. So one, two, three, four, five, six but also two four, six, eight. Any sequences like that tend to have more people choosing them. So you should pick something that feels random to you that you can't see an obvious pattern in it and includes lots of numbers over 30.

Chris - Some good tips for you there, Jess. Do you play the lottery?

Jess - I do not play the lottery, but I'm going to start now!

[Ed - Note from Tim Revell added subsequent to publication in response to enquiries from listeners regarding the fact that one has to discount the order of the numbers selected: "My apologies, I think I missed that bit out on the show when trying not to get bogged down in the mathematical weeds. You take into account the order by getting rid of all of the combinations that are the same i.e. 123456 and 132456. You do that by dividing 59 x 58 x 57 x 56 x 55 x 54 by 6 factorial, which gets you 45057474..."]