The Gamblers' Fallacy

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Randomness. It affects all aspects of life: The chance that a new car turns out to be a lemon. The probability that the next person a salesperson talks to will happen to need what he is selling. Even the chance you have inherited a gene that makes you susceptible to heart disease. I could go on and on of course. Life is uncertain by nature and that’s not going to change in the foreseeable future.

Fortunately we are not entirely helpless in the face of this uncertainty. By informing ourselves we improve our odds on such things as buying cars and sending a salesperson to certain potential customers. While we cannot change our genetic makeup, we can live lifestyles that give us a better chance of healthy, enjoyable lives. However the uncertainty is still there, even if reduced. Fortunately there is a proven method of further reducing that uncertainty, namely the study of probability and statistics. Unfortunately few people understand those subjects at even the most basic level.

For example, suppose you start tossing a coin and it comes up heads the first five times. It is a fair coin, fairly tossed. What is the probability of tails on the next toss? Unless you are educated in statistics, the answer may surprise you – it is 50%, no more and no less. The coin has no memory of what has already happened. No matter how many heads or tails came up before, the next toss of a fair coin always has a 50% probability of heads and 50% of tails.

I know, you may have a hard time believing this. The human mind does not have a natural or intuitive understanding of probability. People need specific education in statistics before they really understand the subject – and even then they often make mistakes. One of those common mistakes is an error that seems built into our minds called the Gamblers’ Fallacy. This is the erroneous belief that some cosmic law forces random events to become non-random in order to compensate for previous results. We tend to think that the coin has a memory and will somehow try to turn up tails if it has previously turned up heads more than expected. Sorry, the universe doesn’t work that way. The coin has no memory, no brain, no way to deliberately compensate for past results. It just lands the way it lands, showing heads or tails independent of anything that happened before.

Still not convinced? Then ask yourself what changed when you got that string of three to five heads with your coin, or when you got three sevens in a row with dice. Is the coin different now than it was a few minutes ago? Are the dice different than they were before? If nothing is different they why do you expect different probabilities? A coin with a 50% chance of landing heads up before you tossed five straight heads will still have that same 50% chance of landing heads up afterwards.

Of course there are some situations in which your action does change the situation. If you have ten red and ten green marbles in a box and pull one out at random, you will have changed the mix in the box. If you draw a red marble, you have a higher chance of a green one next time since the box now has more green than red marbles. In fact should you happen to draw five red marbles and no greens, there will be twice as many green as red marbles left giving you twice the probability of drawing a green as a red on your next try. In this case, drawing the first marbles changed the situation by changing the mix of red and green marbles in the box.

This even has a somewhat practical application. Suppose studies have shown that in a salesperson’s territory 30% of potential customers actually need the product. If he has called on half his potential customers and only 20% of those needed the product, the remaining half of potential customers must include 40% who need what he’s selling. He should be optimistic about calling on the rest. However most salespeople don’t know how many people are likely to need their product. For them, the next customer is just as likely to buy as were the last 20.

The game of blackjack is another example of a changing situation. Under some rules, a clever card counter can adjust his bet according to the modified probabilities for the next deal. The reason is that cards dealt are visible, allowing players to determine what cards are left in the shoe. Years ago a mathematician figured that out and regularly beat the house in Las Vegas. However casino management, not being in the business of giving away money, barred him from playing and changed the rules to ruin his method.

Unfortunately we humans are good at fooling ourselves and the gamblers’ fallacy is one way we do it. In fact, not only do we fool ourselves, there is no shortage of charlatans ready to help us do it. Some are probably honestly misled themselves, while others are deliberately leading us down a path to the loss of our money.

We can even find instructions about something called “regression betting” which claims to tell us how to beat such games as craps. Those instructions are detailed, complicated, and nonsense. They are based on the assumption that you can change the amount you bet to compensate for the changing probabilities based on previous results. That is, they are based on the gamblers’ fallacy that the dice have a memory and will act in a non-random manner to “balance” the overall results. Those instructions may extend the time it takes to lose your money, but they do not change the overall chances of winning or losing. In a game with no house advantage your expectation is still zero. If there is a house advantage your expectation will be negative. The dice have no memory and don’t compensate for anything.

Casino owners love such things as regression betting. They have a saying that if a gambler has a system they will send a taxi for him. They know that such gamblers tend to be so convinced that the system works that they bet more than they would otherwise. The casino makes more money from those people because while their systems don’t, their belief in the system misleads them into thinking that they will win soon. However such systems tend to be cleverly disguised versions of the gamblers’ fallacy, often so cleverly disguised that they fool even the people who concoct them.

“But wait,” you ask. “What about those cases where lottery prizes become so high that they would cover the cost of buying all possible ticket numbers. Isn’t that a situation change deserving of buying a lot of tickets? Doesn’t the expectation become positive then?” This isn’t strictly a gamblers’ fallacy problem but the answer is no, for three reasons:

First, the lottery prize is not a cash prize of the advertised amount. A million dollar prize is really $50,000 per year for 20 years. Meanwhile the lottery owners keep the rest of the money in interest-bearing accounts. If you insist on a cash payout you will get much less than a million bucks.

Second, even if you win there is no guarantee you will be the only winner. When prizes get large, lots of people play so the prize will probably be split between two, three, or even four winners.

Third, should you win you will have to pay taxes on your winnings, and those taxes will probably be in a high tax bracket.

Even when the prize is huge, lotteries provide negative expectation for the price of the ticket.

So whether buying a car, trying to sell a product, or playing the lottery, don’t believe that the odds will ever change. They don’t unless the situation changes.

Next time I intend to discuss some other ways our minds fool us in situations of uncertainty.