Wrote for Luck

Fans of probability love random processes. And lotteries are a great example of random number generation.

The UK National Lottery ran in one format from 19/11/1994 until 7/10/2015. I was talking to somebody who had played the same set of numbers in all of these lottery draws and I wondered what the net gain or loss has been for them over this period.

The basic format is that people buy a line of numbers (6 numbers, from 1-49) and try to match the six numbers (from 49 balls numbered 1-49) drawn from a machine. The aim is to match all six balls and win the jackpot. The odds of this are fantastically small (1 in ~14 million), but if they are the only person matching these numbers they can take away £3-5 million. There are prizes for matching three numbers (1 in ~56 chance), four numbers (1 in ~1,032),  five numbers (1 in ~55,491) or five numbers plus a seventh “bonus ball” (1 in ~2,330,636). Typical prizes are £10, £100, £1,500, or £50,000, respectively.

The data for all draws are available here. I pulled all draws regardless of machine that was used or what set of balls was used. This is what the data look like.

The rows are the seven balls (colour coded 1-49) that came out of the machine over 2065 draws.

I wrote a quick bit of code which generated all possible combinations of lottery numbers and compared all of these combinations to the real-life draws. The 1 in 14 million that I referred to earlier is actually

\(^{n}C_r = ^{49}C_6\)

\(\frac{49!}{\left ( 6! \left (49-6 \right )! \right )} = 13,983,816\)

This gives us the following.

Crunching these combinations against the real-life draw outcomes tells us what would have happened if every possible ticket had been bought for all draws. If we assume a £1 stake for each draw and ~14 million people each buying a unique combination line. Each person has staked £2065 for the draws we are considering.

• The unluckiest line is 6, 7, 10, 21, 26, 36. This would’ve only won 12 lots of three balls, i.e. £120 – a net loss of £1945
• The luckiest line is 3, 6, 13, 23, 27, 49. These numbers won 41 x three ball, 2 x four ball, 1 x jackpot, 1 x 5 balls + bonus.
• Out of all possible combinations, 13728621 of them are in the red by anything from £5 to £1945. This is 98.2% of combinations.

Pretty terrible odds all-in-all. Note that I used the typical payout values for this calculation. If all possible tickets had been purchased the payouts would be much higher. So this calculation tells us what an individual could expect if they played the same numbers for every draw.

Note that the unluckiest line and the luckiest line have an equal probability of success in the 2066th draw. There is nothing intrinsically unlucky or lucky about these numbers!

I played the lottery a few times when it started with a specified set of numbers. I matched 3 balls twice and 4 balls once. I’ve not played since 1998 or so. Using another function in my code, I could check what would’ve happened if I’d kept playing all those intervening years. Fortunately, I would’ve looked forward to a net loss with 43 x three balls and 2 x four balls. Since I actually had a ticket for some of those wins and hardly any for the 2020 losing draws, I feel OK about that. Discovering that my line had actually matched the jackpot would’ve been weird, so I’m glad that wasn’t the case.

There’s lots of fun to be had with this dataset and a quick google suggests that there are plenty of sites on the web doing just that.

Here’s a quick plot for fun. The frequency of balls drawn in the dataset:

• The ball drawn the least is 13
• The one drawn the most is 38
• Expected number of appearances is 295 (14455/49).
• 14455 is 7 balls from 2065 draws

Since October 2015, the Lottery changed to 1-59 balls and so the dataset used here is effectively complete unless they revert to the old format.

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The title of this post comes from “Wrote for Luck” by The Happy Mondays from their 1988 LP Bummed. The Manic Street Preachers recorded a great cover version which was on the B-Side of Roses in The Hospital single.