10 mathematical reasons you're not a millionaire… yet

Imagine you stumble across a way to buy a supply of magic beans on Mondays. On Friday afternoon you can always sell the beans you bought for twice the price you paid for them. So you spend your start-up money on a supply of magic beans, sell them and double your money, then use the money to stock up on twice as many magic beans the next week. Hey presto, you can then keep on doubling your money – after one week you will have £2,000, after two weeks £4,000 and so on until after 10 weeks you will have £1,024,000. Bingo!

Of course, this isn’t really a practical plan. It does however demonstrate the magic of exponential growth: always bear in mind if you have £1,000 in the bank and earn £100 a week you won’t get rich as quickly as you would if you instead found a way to earn 10 per cent compound interest a week (where the interest is paid on the initial sum and the interest earned). After just a year and a half, the latter would have earned you more than twice as much as the former.

When considering an investment opportunity or business model, it is often useful to know how long it will take to double your money at a particular rate of growth. The Rule of 72 is a quick way to calculate this in your head. It has been used since at least the 15th Century when Luca Pacioli (1445–1514) included it in his Summa de arithmetica.

The rule is to divide 72 by the rate of growth (or the interest rate, for savings and investments): the result gives you the number of periods it will take for the initial investment to be doubled. For instance, for an interest rate of 9 per cent a year, we divide 72 by 9 and get eight years. The actual time it would take money to double at 9 per cent is 8.043 years, so this is reasonably accurate.

It’s an old joke that the best way to make small fortune from gambling is to start out with a large fortune. But joking aside, some of the best ways to make your money grow at an exponential rate are only really available to those with significant wealth already at their disposal.

Land prices in most economies and the major markets for stocks or shares have tended to increase at about 5-10 per cent over inflation for decades. For instance, investing in index funds (which track the performance of the entire market) will generally give this kind of return. Not exactly magic beans, but a pretty good substitute for those with sufficient funds. The catch is that you need to have enough money to lock it up for long periods in such investments, an opportunity that is most readily available to the wealthiest.

Some of the best-known examples of hedge funds and investment funds are the successful ones, but there have been plenty that went out of business after their leveraged gambling led to massive losses. Nassim Nicholas Taleb has written well about how ‘survivorship bias’ makes us overestimate how good investment managers are (and how highly they rate themselves as well).

Even if investment were a completely random process, there would be some successes and some failures, and the successful managers would no doubt believe themselves to be brilliant. More importantly, it is worth bearing in mind that managed funds have consistently been outperformed by index funds, which means that, on average, fund managers do worse than you could do by picking stocks at random.

Casinos and bookies wouldn’t make money if the odds weren’t loaded in their favour. The house edge is defined as the casino’s expected profit, expressed as a percentage of the original bet. For most games the way to calculate this is to analyse the range of probabilities across the sample space for that game.

For instance, when playing roulette with a single zero, there are 37 possible outcomes. If you bet £1 on red, there are 18 winning slots and 19 losing slots on the roulette wheel, each of which is equally probable. So your expected value is 18/37 – 19/37 = -1/37

Converted into percentage terms, this is an expected value of –2.7 per cent, meaning that the house has a positive expected value (or edge) of 2.7%. And because of the law of large numbers, over a large number of tables and games, the casino can expect the luck to more or less even out, leaving them with something very close to this as their actual profit. (Incidentally, ‘luck’ is pretty much the same thing as the statistical concept of ‘standard deviation’, but this article is too short to fully explain that connection…)

The basic principle behind hedge funds comes from the world of gambling: ‘arbitrage’ is the practise of placing bets with different bookmakers so as to take advantage of differential odds, while ‘hedging’ is more often used by gamblers to exploit variations of odds over time in the same market.

For instance, let’s say a bookmaker takes your bet of $100 at 4/5 on (digital odds of 1.80) on the New York Giants to beat the Denver Broncos at the Superbowl, at that same time as he is also offering bets on the Broncos at the same price, giving him an edge of 10 per cent in total.

But now let’s say that the Broncos star quarterback gets injured in the warm-up, or the Giants take an early lead during the game. Now you can get the Broncos at odds of 3/2 (digital odds: 2.50). So you can nail down a guaranteed profit, whichever team wins, if you get your sums right (and if the change in the game is in your favour).

A good way to understand the underlying flaw in all gambling systems is to examine the maths of the notorious Martingale system. This involves starting with a single unit bet on an evens bet such as red or black at the roulette table. If you win, you bank the win and start again. If you lose, you double your bet for the next round.

For any completed cycle you come away with 1 unit winnings: for instance, 8 units returned for 1 + 2 + 4 = 7 units on the third round, 16 units returned for 1 + 2 + 4 + 8 = 15 units staked and so on.

Of course the problem is that your stake is increasing exponentially, so all it takes is a run of losses larger than your initial bankroll to wipe you out. For instance 10 losses in a row would add up to 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 = 1023 units staked. If you started with 1024 units then you have only 1 unit left, which you will need to hang on to, so you can buy a drink to drown your sorrows.

Like most systems, the Martingale concentrates all the risk into one relatively infrequent event, which is why it has bankrupted so many gullible gamblers.

Bookmakers are pretty smart and spend a lot of time analysing statistics in order to avoid obvious mistakes. But they can still slip up.

In 1991, Paul Simmons and John Carter, from Essex in the UK spotted a potential moneyspinner. Many bookmakers treated bets on holes-in-one at golf tournaments as a novelty long-odds bet, offering anything from 3/1 to 100/1. However, a close study of the exact statistics convinced them that the actual odds for any player to score a hole in one at a particular tournament were better than even, so the edge was, for once, in the gamblers’ favour.

This was the pre-internet era, so they embarked on a whirlwind road trip of towns around Britain, placing bet after bet at smaller bookies. When their bets all won, at least one bookmaker closed down and left the country rather than pay, but most paid up and they made several hundred thousand pounds.

The moral of the story is that where people are pricing risk based on their intuition rather than the statistics, there will often be an opportunity to profit from the resulting errors.

In most gambling situations, the odds are rigged so that you can’t simply bet on every option and guarantee a profit. However, when it comes to lotteries there are occasionally situations like rollovers where it is theoretically possible to buy every single ticket and be sure of winning.

In the early 1960s, the Romanian economist Stefan Mandel saw an opportunity to do this in his country’s lottery. He couldn’t raise the finances to buy every single ticket – so instead he manually worked out the combinations he required to guarantee getting at least five of the six numbers. Luckily, he managed to hit the jackpot first time and raised enough money to migrate with his family to Australia.

There, he organised a syndicate which could raise enough money to fund further attempts: it won the lottery 12 more times in Australia before he moved to live in luxury on a small South Pacific island.

Many cons rely on our failure to understand basic probability. For instance, in the ‘perfect prediction scam’, the scammer picks an event that has a binary outcome, such as a knockout soccer match, or whether the stock market will rise or fall in a given week. They then send out 16,000 emails claiming that the scammer has a perfect prediction method or inside information, half of which predict winner A and half winner B. The following week they follow up on the 8,000 that had the winner with a new prediction, again, split half and half.

After four rounds of this they end up with 1,000 people who have received four correct predictions in a row. Now the scammer offers to continue sending predictions to these people, in return for a fee of, say $100. If even 10 per cent of the 1,000 accept the offer, the scammer has made $10,000 (minus the costs of setting up the scam.) Which really is money for nothing!

WARNING: Gambling can be addictive. More information and support for anyone affected directly or indirectly by problem gambling can be found on the BBC addiction information and support page.

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