A Maserati for £1

After Dave and Angela Dawes won £101 million on the Euromillions lottery, Radio 5 Live asked me to comment on the different ways one could win a decent amount of money for £1. I chose £100,000, which will buy you a shiny new Maserati ( a Ferrari would be about double that). The recording of my interview is here, and here are the details of my calculations, which I hope are roughly correct.

National Lottery

There are 49 balls and if your choice of 6 numbers matches 5 winning balls plus the bonus number (a 7th ball drawn) then this generally wins around £100,000 and has a probability 1 in 2,330,636.

‘Health’ Lottery

The new ‘Health Lottery’ has a poor payout to the punters of around 33p in the pound, but 5 numbers out of 50 would win ‘up to £100,000’ with probability 1 in 2,118,760.

Premium bonds

There are around 41,000,000,000 Premium bonds, and currently each month 4 win £100,000 and 1 wins £1,000,000. So with a £1 Bond there is around 1 in 8,000,000,000 of winning at least £100,000.

However you keep your £1 stake, and so a fairer comparison is to assume that you find a friend to lend you £500 for a month for £1 – this is 2.4% annual interest which is not too bad at the moment. With 500 bonds held for a month the odds of wining at least £100,000 are 1 in 16,000,000, around 8 times worse than the lotteries.

Horse-racing accumulator

Suppose you examine a race meeting with 6 races, and in each race choose a horse at medium odds of around 6 to 1 against. Then an accumulator, in which the winnings of each race are passed to the next horse, will pay out, will give you $7 \times 7 \times 7 \times 7 \times 7 \times 7$ = £117,000 if they all win. Given a bookmakers margin of, say, 15% each bet, the true odds may be around 1 in 230,000.

Roulette

If you can find a casino to let you bet £1, place it on your lucky number between 1 and 36. When it wins, either leave the £36 there or move it to another number. When that comes up too, move the £1296 you now have to another number, or leave it where it is – it doesn’t make any difference to the odds, but somehow it seems that the chance increases when the money is moved.

When that comes up you will have £46.656, so move it all to Red, and when that comes up you will have £93,312, almost enough for your Maserati.

The chances of this happening, on a European roulette wheel with 1 zero, are
$ 1/37 \times 1/37 \times 1/37 \times 18/37 $= 1 in 104,120.

Conclusion

Roulette is your best bet – about twice as good as horse-racing, about 20 times as good as lotteries, and about 160 times as good as premium bonds.

Comments

Philip Potter 's picture

Nice analysis. I can't help but feel you aren't comparing like with like entirely - for example, the races and roulette strategies will win you either £100000 or nothing at all, while the lotteries and premium bonds have a chance for paying out some intermediate amount. You could, for example, buy a lottery ticket, and win £10 which you could then sink into further lottery tickets to boost your chances of getting that maserati...
david's picture

yes, I was doing an 'all or nothing' analysis. But in fact the additional chance from re-investing one's lottery winnings is fairly feeble, given that there is only a 1 in 52 chance of winning £10.

Ewan Leeming's picture

I find it fascinating that if you were to rank those methods of gambling in order of social acceptability, it would be almost completely backwards.
Dave Marsay's picture

My first thought was that Dave S must have his Maserati now, and hence had a very sharp discounting of value with time. But then I noticed that he would have to wait a month for his premium bonds, in which time he could have re-gambled any small lottery winnings. So I imagine he is both impatient and lazy. ;-)
Stavros Christofides's picture

I fully support the need to explain uncertainty, probability and odds to anyone that will listen. Suggesting that one can win a £100,000 Maserati for £1 is catching even if statistically you know ‘it’s not going to be you’! The comparisons are, as other have pointed out, somewhat simplistic and misleading. If we had a number of games that would accept a £1 for a one and only possible win of £100,000 it would simply be a matter of comparing the actual odds behind each game. Once we have a number of options things get more complicated and we need a way of assessing the implicit risk loadings behind these offerings in order to compare them. There is a sound way to rank these different games of chance by deriving the ‘risk aversion level’ (RAL) implicit in their cost and expected returns. This approach is based on an actuarial pricing principle, the so-called Wang Proportional Hazards transform which has been shown to uniquely satisfy a demanding set of pricing axioms. The method transforms the underlying Survival function (= 1-Cumulative Distribution Function), by raising this to a power of one or less. It is helpful here to recall that the area under the Survival function is the mean or expected value and the mean of the transformed distribution then gives the price. An Index of 1 leaves the distribution and the mean unchanged and as the index decreases to zero the price increases from the expected gain (or loss) to the maximum gain or loss. These two conditions are known as ‘positive loading’ and ‘no rip-off’. Let us now look at a simple game of Roulette to see how these calculations work. We start with a bet on just the colour in a simple game of (European) Roulette where the gambler bets on one of two colours, with an expected probability of winning back twice his stake of 18/37. The mean outcome, for a £1 stake, is then 36/37, slightly below his stake. The survival function is very simple in this instance as there are only two possible outcomes, zero, with probability 19/37 and 2 with probability 18/37. The ‘risk adjusted’ cost at a RAL of say z, is then simply the solution to the equation: 2 * (18/37)^(z ) = 1 which gives z = ln( 2) / ( ln (37) - ln (18)) = 0.961975 Note also that the price according to this principle ranges from 18/37 whenz=1 and increases asymptotically to 2 as we increase z. Betting on a single number changes the odds and the z (RAL). Here the gambler bets £1 and collects £36 with probability 1/37. The ‘risk adjusted’ cost at RAL z is then simply the solution to the equation: 36 * (1/37)^(z ) = 1 which gives z = ln (36) / ln( 37) = 0.992412 To play to win £99,312 requires three number bets and a colour bet, with the odds becoming 1 in 104,120. The z in this case is given by z= ln(99312)/ln(104120) = 0.99051. The higher value of z makes this option the more attractive of the two examples. In the case of the Lotto, a number of combinations can win from £10 to many millions of pounds. The expected win (or payout ratio) is around 45% and the implicit RAL of such a game is then the value of the z that will produce an adjusted mean of £1. These calculations are best done in a table and this can be used to extract the parameter by using Solver in Excel. For more details on this and the other examples below see the link at the end of this response. The standard Lotto RAL turns out to 0.925793. Carrying out post-draw analyses of actual results produces values from around 0.917 to 0.935 depending on whether or not a jackpot has been won. These values can change dramatically in the case of a ‘rollover’ draw when they can as be around 0.961538. When this happens it is clearly a time for people who do not normally participate to take a punt! The situation with National Savings Certificates is somewhat different as the cost of each monthly draw is the loss of interest that the bonds invested would generate during the month. From around an annual rate of 5% back in 1998 we are now down to 1.5% which has significantly reduced the value of prizes with the low prize now down to £25 from £50. The challenge here is to actually work out what has been paid or given up in order to participate in the monthly draw. As Bond prizes are tax free this is further complicated by tax. The actual cost is then the net rate available to the investor with the same level of security, which is currently around 2.5%. We can now estimate the RAL underpinning the Premium Bond pricing for such an investor using a table approach as was suggested for the Lottery. This turns out to be a RAL z of 0.955566. This is better than Lotto, except in a ‘rollover’ week. The clear winner here is then playing numbers on the Roulette wheel. The trick is not to get addicted. http://www.actuaries.org.uk/research-and-resources/documents/pricing-risk-financial-transactions
Tom's picture

You can heavily reduce the bookmaker's margin on any sports betting accumulator without any knowledge of the sport by simply using an odds comparison site and only picking selections where your chosen bookmaker is offering standout odds. Therefore a sports bet accumulator is the best chance
Jon's picture

Not serious this really but: I was reading this blog entry and a new email alert popped up on my screen, titled "Fancy a free bet this weekend?"... so I'm off to do a sports accumulator bet, whatever that is...