I thought it would be a good idea to look at the ROI (Return on Investment) for lotteries, to get an idea of just how bad they are, on average, as an investment. Some lottery prizes are fixed amounts, whereas for others we can work out an ROI based on average payouts.
The English lottery is fairly simple, and a good place to start. If we ignore the bonus ball, we see these odds (as of September 2018):
- Match 2: 1 in 10.3 - Free Lotto Lucky Dip
- Match 3: 1 in 97 - Set prize of £25
- Match 4: 1 in 2,180 - £100
- Match 5: 1 in 144,415 - £1,000
- Match 6: 1 in 45,057,474 - Jackpot
A free Lucky Dip ticket is worth £2, and as of 2016 the maximum jackpot payout is £22 million. We will take two thirds of this as our average jackpot win, which is quite possibly a bit overly generous.
ROI is a measure of the amount of profit from an investment, as a percentage of the initial investment amount. This gives us the following ROI for each win:
- Match 2: (2 / 10.3) - 1 = -0.8058
- Match 3: (25 / 97) - 1 = -0.7423
- Match 4: (100 / 2,180) - 1 = -0.9541
- Match 5: (1,000 / 144,415) - 1 = -0.9931
- Match 6: (2,000,000 x 2/3 / 45,057,474) - 1 = -0.6745
Notice that all of these ROIs are negative - this is because you will not only not make a profit, on average, when playing the lottery, you will also lose most of your initial investment. For some of the prize levels, the disparity is huge - a small payout, with very long odds against winning.
Of course, there's an easier way to work this out. Lotteries often published the details of their methods, including a payout percentage. For the UK lottery, 53% goes on prize money - so an average ROI would be -0.47. For every pound you invest in the lottery, you will lose 47p of your initial investment come the day of the draw.
The ROI can be affected when people are allowed to choose their own lottery numbers. For one UK lottery draw, where there was a guaranteed payout for 3 numbers and a percentage for 5 numbers, the people with 3 numbers made more money because so many people had chosen 5 of the 6 numbers correctly - the winning numbers were 7, 14, 21, 35, 41 and 42 (and bonus ball 43). It turned out that people like patterns, and of the 6 numbers in the draw, 5 of them were multiples of 7. Such a large number of people had chosen multiples of 7 (presumably 7, 14, 21, 28, 35 and 42 - with 28 being the ball that wasn't picked) for their numbers that, by the time the prize money was split up and shared between them, it was just over a half (£15) of the fixed amount (£25) given out to the 3 number winners.