by Matt Parker
You Me Nothing to see here
HHH THH 12.5%
HHT THH 25%
HTH HHT 33.3%
HTT HHT 33.3%
THH TTH 33.3%
THT TTH 33.3%
TTH HTT 25%
TTT HTT 12.5%
You’ll see a list of percentages over to the right. There is no need to bother yourself with those. They are the just the probabilities that you will win. You may have noticed that they are all below 50 per cent. Yet, as long as you choose first, I can always make a prediction which has a better chance of winning. The best-case scenario for me is when my opponent always goes for HHH or TTT, which gives them a 12.5 per cent chance of winning and me an 87.5 per cent chance of success. Even if I assume my opponent chooses their sequence at random, I have a 74 per cent chance of winning on average.
When I first saw this game, it did not make sense to my poor brain. Each coin flip was independent, yet something strange was going on with predictions of three flips in a row. The sneakiness is in how the coin is continually flipped until one of the predicted runs of three occurs. If the coin was flipped three times to get one result, then flipped a whole new three times for the next result, then the outcomes would be independent. But if the last two flips of one run of three form the first two flips of the next result, the results overlap each other, then they are no longer independent.
Take a look at my predictions: the final two choices of mine are the same as the first two of my opponent. My goal is to cut them off at the pass. Sure, one player could win on the very first three flips of the coin (a 12.5 per cent chance for each person), but after that the winning run of heads and tails will be preceded by a run of three which overlaps it. I want to choose that preceding group. For something like TTT it either has to be the first three flips or it will definitely be beaten by HTT coming directly before it. A mid-sequence run of three tails will always have a head directly before it, giving HTT before TTT. The game is rigged. This is a game known as Penney Ante and it has been used to separate humans from their cash for years.
In the game of Penney Ante people get thrown because every option of three heads and tails has a different combination which is more likely to win. It is unsettling that there is no best option to pick which is more likely to win than all the others. But this exact oddity is the basis of the game Rock, Paper, Scissors. Any option picked can be beaten by one of the other options.
This is the difference between transitive and non-transitive relations. A transitive relation is one that can be passed along a chain. The size of real numbers is transitive: if nine is bigger than eight and eight is bigger than seven, then we can assume that nine is bigger than seven. Winning in Rock, Paper, Scissors is non-transitive. Scissors beats paper and paper beats rock but that does not imply that scissors can beat rock.
The second probability game you can use to trick humans was invented by mathematician James Grime. He developed a set of non-transitive dice which now bear his name: Grime Dice (which is now the second-best boy-band name in this book). They come in five colours (red, blue, green, yellow and magenta) and you can use them to play a game of Highest Number Wins. You and your opponent choose a dice each and roll them at the same time to see who gets the highest number. But for every dice there is a different-coloured dice which will beat it more often than not.
On average: red beats blue; blue beats green; green beats yellow; yellow beats magenta; and then magenta beats red. My contribution to the dice was to wait for James to work the numbers out and then suggest a range of colours to help remember the red-blue-green-yellow-magenta order; each colour is one letter longer than the previous one. Now you let your opponent choose their colour first and you pick the dice with one fewer letters (red (3) rolls over to magenta (7)).
If you use these dice to win drinks and money off your friends and family, you may eventually have to come clean about their non-transitiveness. Maybe even teach them the order of the dice. But then suggest doubling the dice up and rolling two of each colour together. Because when the dice are rolled in pairs it perfectly reverses the order of which beats which. Instead of red beating blue, blue now beats red more often than not, and so on. If you go first, your opponent will still be using the single-dice system and select a colour which is likely to lose to yours.
Non-transitive dice are a relative newcomer to the world of mathematics. They appeared on the maths scene only in the 1970s, but they quickly made a big impact. Multibillionaire investor Warren Buffett is a big fan of non-transitive dice and brought them out when he met also-multibillionaire computer guy Bill Gates. The story goes that Gates’s suspicion was aroused when Buffett insisted he pick his dice first and, upon a closer inspection of the numbers, he in turn insisted Buffett choose first. The link between people who like non-transitive dice and billionaires may be only correlation and not causation.
James Grime’s contribution to the non-transitive world was to make it so that his dice have two different possible cycles of non-transitiveness but with only one of them reversing when you double the dice.fn1 By renaming the green dice ‘olive’, the second cycle can be remembered as the alphabetical order of the colours. Using both cycles, in theory, you can let two other people choose their dice colours and, as long as you can then choose the one- or two-dice version of the game, you can beat both opponents simultaneously more often than not.
I wish I could say there is some amazing mathematics going on behind the scenes which makes Grime Dice work, but there isn’t. James decided the properties he wanted the dice to have and then spent ages working out what numbers would allow that to happen. If you were to give me two different six-sided dice with whatever random numbers from zero to nine you want on them, I can find a third dice to complete a non-transitive loop more than one time in three. The maths is only amazing because it catches the human brain off guard. But be warned: humans brains are quick to hold a grudge if you win too many drinks off them.
You’ve got to be in it to not win it
There is nothing you can do to increase your chances of winning the lottery other than buy more tickets. Wait – I should specify: buy more tickets with different numbers. If you buy multiple tickets with identical numbers, then you don’t increase your chances of winning. But if you do win with multiple tickets and have to share the prize, you’ll get a bigger portion. So it’s a way to win more money, but not to win more often.
But surely no one has ever won the lottery with multiple identical tickets … Except Derek Ladner, who in 2006 accidentally bought his ticket for the UK lottery twice. Three other people also won, so instead of getting a quarter of the £2.5 million jackpot, he took home two-fifths. He had claimed the first fifth before realizing he had one of the other winning tickets … And Mary Wollens, who deliberately bought two identical tickets for a Canadian lottery (also in 2006) and took home two-thirds of the $24 million instead of half … And the husband and wife in 2014 who each bought their regular UK lottery ticket without telling the other (they matched five out of six numbers and the bonus ball) … And Kenneth Stokes in Massachusetts, who played his regular numbers on the Lucky for Life lottery, despite his family buying him an annual ticket.
But if you want to increase your chances of winning, you need to buy two different tickets. Not that it’s a financially smart decision. On average, every time you buy a lottery ticket you lose money. The current licence issued by the UK Gambling Commission to Camelot UK Lotteries Limited stipulates that 47.5 per cent of the money spent on lottery tickets needs to be given back as prizes (on average, actual prizes fluctuate week to week). This is the expected return in black and white. For every £1 a player spends on a lottery ticket, they can expect to get 47.5p back in prizes.
But people do not gamble because of the expected return. Running a lottery is actually about skewing the distribution of prizes as far from the expected return as is feasible. I could put in a competing bid for the national lottery licence and undercut Camelot by having dramatically lo
wer admin costs. My plan is that, when people buy a £2 ticket, the person at the point of sale just gives them their expected 95p prize there and then. It cuts back on admin and I wouldn’t even need to bother drawing the numbers twice a week.
It’s a ridiculous and extreme example, but it gets the point across: people do not want their expected return, they want a chance to get back more than they put in. All right, so now every third customer gets £2.85 back and everyone else gets nothing. Or every fourth ticket pays out £3.80. When is it skewed enough? Should every hundredth customer get £95? There are things like scratch cards which operate around this value of prize (and in fact have higher expected returns), but the lottery has decided to really skew us.
In 2015 Camelot made it harder to win the lottery. Instead of choosing six numbers from a total of forty-nine numbers, they changed it to choosing six numbers from fifty-nine. In one of my favourite bits of PR spin ever, they sold it as ‘more numbers to pick from’. Ha! The reality is that there were now more numbers a player wasn’t going to choose, dramatically lengthening the odds. Picking six numbers from forty-nine gave a one in 13,983,816 probability of winning the top prize, whereas choosing six from fifty-nine now gives a one in 45,057,474 chance. If you factor in that one of the new, lower prizes was a free ticket in the next draw (I did), the odds of winning per ticket purchase was one in 40,665,099. At the time, I described it as being more likely that a UK citizen picked at random would have Prince Charles as their dad. It’s exceedingly unlikely.
However, despite all this, I would argue that the new reduced odds of winning actually made the lottery better value. The average payout had not changed, they had just decided to hand it out in bigger lumps to fewer people. The rule changes have resulted in more jackpot rollovers, which make for much bigger prizes – the sorts of prizes that get media attention. And people are not buying tickets for the expected value, they are buying the permission to dream. Having a non-zero chance of winning a life-changing amount of money allows someone to dream about that version of their life. The more publicity a lottery draw gets and the more life-changing the prizes are, the bigger those dreams can be. Which is, arguably, better value.
A load of balls
There are people online trying to sell their secrets to winning the lottery. Much of the pseudoscience around lottery draws tries to cloak itself as being mathematical and is normally a variation on the gambler’s fallacy. This logical fallacy is that, if some random event has not happened for a while, then it is ‘due’. But if events are truly random and independent, then an outcome cannot be more or less likely, based on what has come before it. Yet people track which numbers have not come up in the lottery recently to see which ones are due an appearance.
This reached fever pitch in Italy in 2005 when the number 53 had not been seen for a very long time. The Italian lottery in 2005 was a bit different to that in other countries: they had ten different draws (named after different cities), in each of which participants chose five numbers from a possible ninety. Unusually, players don’t have to choose a complete set of numbers. They can opt to bet on a single number coming out of a certain draw. And the number 53 had not come out of the Venice draw for nearly two years.
Loads of people felt that the Venice 53 ball was due. At least €3.5 billion was spent buying tickets with the number 53; that’s €227 per family in Italy. People were borrowing money to place bets as 53 continued not to get drawn and so was, apparently, more and more overdue. Those with a system kept increasing their stake each week so when 53 finally arrived they would recoup all previous losses. Players were going bankrupt and, in the lead-up to 53 finally being drawn, on 9 February 2005, four people died (one lone suicide and a second suicide who took their family’s lives as well).
Italy even has a cult-like collection of people who believe that no number can take longer than 220 draws before coming out. They call this the ‘maximum delay’ (or rather, the ritardo massimo) and base it on the early-twentieth-century writings of Samaritani. The mathematician Adam Atkinson, with other Italian academics, was able to reverse-engineer the Samaritani Formula to show that Samaritani had worked out a good estimate of what the expected longest run should be between draws of any given number (for the lottery at the time). Somehow, this estimate transformed over the generations into a supposed magical hard limit for any lottery.
Another thing that happens is that people mistakenly think that recent results are unlikely to happen again. I’ve seen advice online like ‘Don’t choose numbers which have won the big jackpot before’ and ‘Using a combination that has gone through the system already will stack the odds even higher,’ and it is all rubbish.
In 2009 the Bulgarian lottery drew the same numbers – 4, 15, 23, 24, 35 and 42 – two draws in a row, on 6 and 10 September. They were drawn in a different order but, in a lottery, the order does not matter. Amazingly, no one won the jackpot the first time they were drawn but, the following week, eighteen people had chosen them, in the hope they would come up again. The Bulgarian authorities launched an investigation to check nothing untoward was going on, but the lottery organizers said that it was just random probability. And they were right.
The only legitimate mathematical strategy you have is to choose numbers that other people are less likely to have also picked. Humans are not very creative at choosing their numbers. On 23 March 2016 the winning UK lottery numbers were 7, 14, 21, 35, 41 and 42. Only one off from a run of all multiples of seven. An incredible 4,082 people matched five numbers that week (presumably, the five multiples of seven; Camelot don’t release that data), so the prize money had to be shared between about eighty times more people than normal: they got only £15 each (less than the £25 people with three balls correct received!). It is believed that, in the UK, around ten thousand people all choose 1, 2, 3, 4, 5 and 6 every week. If they do ever come up, the winners will not get much each. They will not even get a unique funny story to tell.
Top tips are to choose numbers which are not in an obvious sequence, aren’t likely to be numbers from dates (people choose birthdays, anniversaries, and so on) and don’t conform to any misguided expectations of which numbers are ‘due’. Then, if you play the lottery weekly for millions of years (you’d expect to win the UK lottery once every 780,000 years), on the occasions you do win you will have to share the prize less, on average. Sadly, it’s not a strategy that helps much on the timescale of a human lifetime.
So, toppest tip is, if you do play the lottery, just choose whatever numbers you want. I think the only advantage of choosing really random numbers with high entropy is that they look like the winning numbers most weeks – which helps keep the illusion alive that you could have won. And, at the end of the day, that illusion of maybe winning is what you are really buying.
Probably in conclusion
I have an uneasy relationship with probability. There is no other area of mathematics where I am as uncertain about my calculations as I am when I’m working out the chance of something happening. Even for something which has a calculable probability, like the chance of a complicated poker hand, I’m still always worried that I’ve missed thinking about a certain case or nuance. To be honest, I’d be a lot better at poker if I looked up from my calculations and noticed the other players; they could be sweating profusely and I’d not notice, as I’m too busy trying to estimate what ‘52 choose 5’ is.
And probability is an area of maths where not only does our intuition fail us, it is also generally wrong. We’ve evolved to jump to probabilistic conclusions which give us the greatest chance of survival, not the most accurate result. In my imaginary cartoon version of human evolution, the false positives of assuming there is a danger when there isn’t are usually not punished as severely as when a human underestimates a risk and gets eaten. The selection pressure is not on accuracy. Wrong and alive is evolutionarily better than correct and dead.
But we owe it to ourselves to try to work out these probabilities as best we can. This is what R
ichard Feynman was faced with during the investigation into the shuttle disaster. The managers and high-up people in NASA were saying that each shuttle launch had only a one in 100,000 chance of disaster. But, to Feynman’s ears, that did not sound right. He realized it would mean there could be a shuttle launch every day for three hundred years with only one disaster.
Almost nothing is that safe. In 1986, the same year as the disaster, there were 46,087 deaths on roads in the US – but Americans drove a total of 1,838,240,000,000 miles in that year. Which means a journey of around 400 miles had a one in one hundred thousand chance of ending in a fatal disaster (for comparison, in 2015 it was 882 miles). The shuttle was cutting-edge space travel, which is always going to be more dangerous than driving 400 miles in a car. The odds of one in one hundred thousand was not a sensible estimate of the probability.
When Feynman asked the actual engineers and people working on the space shuttle what they thought the chance of disaster was on any given flight, they gave answers of around one in fifty to one in three hundred. This is very different to what the manufacturers (one in ten thousand) and NASA management (one in one hundred thousand) believed. In hindsight, we now know that, of the 135 flights (before the shuttle programme was ended in 2011), two of them ended in disaster. A rate of one in 67.5.
