»

# The Heads and Tails of Probability by Tom Hagen and Sonia Weiss

Gambling on games is based on probability, the branch of mathematics that focuses on calculating the likelihood of a specific occurrence, such as a coin landing face up or a roulette wheel landing on black. Probability is expressed as a number between 1 and 0. Something that is certain to happen — such as the fact that a coin will land face up or face down — has a probability of 1. An event with a probability of 0 is considered an impossibility, while one with a probability of .5 has — you guessed it — equal odds of occurring or not.

## Simple Probability

To illustrate probability, let's take a very simple example — tossing a coin. Coin tossing is easy to analyze, since there are only two ways a coin can land — heads or tails — when it's tossed. No matter whether you call heads or tails, there is one chance out of two that you will win the toss. Expressed another way, there is a 1 in 2 chance that the coin will land in your favor. This is a probability of ½, or .5.

A coin toss is known as an independent event, because the outcome of one toss has nothing to do with what will happen on subsequent tosses. (A dependent event, as the term implies, has outcomes based on other factors, such as what happened before the event.) Let's say you were to toss a coin 100 times, and you're going to bet \$1 on each toss that the coin will land on heads. If it does, you win a dollar plus your bet, doubling your money. If it doesn't, you win nothing and forfeit your bet. Will you come out ahead? Let's do the math.

You know there is a 50/50 chance, or a .5 probability, that the coin will land heads-up on each toss. Every time it does, you win \$2. \$2 times 50 is \$100. However, you are also forfeiting \$1 on every toss you don't win. On a .5 probability, out of every \$100 you put up, you stand the chance to lose fifty tosses, or \$50. Theoretically, you would not lose or gain anything on your original bankroll. You'd end up even.

Flipping a coin to determine who gets the ball first or to settle a dispute really isn't all that fair. Human-generated flips introduce factors that affect probability, as the coin isn't launched the same way twice. Researchers have also found that when a tossed coin starts out heads, it ends up heads when caught more often than tails.

We say theoretically because probability theory tells us that this is what should happen. Does this mean that in 100 coin tosses, exactly 50 percent of those will be heads? Definitely not. But the odds are that they will even out, over time, based on the theory of the law of large numbers. This theory states that a large sample of a particular probabilistic event will tend to reflect the underlying probabilities. In layman's terms, what this means is that the more times you flip a coin, the closer you get to coming up with 50 percent heads and 50 percent tails, or hitting 50/50 odds.

## Going Against the Odds

Sometimes probability theory doesn't seem like it really works. If you toss that coin and it lands heads-up a number of times in succession, it may seem like you are going against the odds. You might be for a short time, but you won't be over the long haul. The .5 probability will hold. In other words, the odds remain at 50/50 since a coin — or mathematics, for that matter — has no memory. The next toss could result in heads. Then again, it might not.

Unfortunately, this kind of thinking gets some gamblers into big trouble. They get on a winning streak and start betting big because they believe the streak will continue. Or the opposite happens. They get on a losing streak and start betting big because they think the odds favor an end to the streak. They might, but it could happen long after they lose all their money.

Thinking that a run of luck — good or bad — will influence the odds of winning or losing in the future is called the gambler's fallacy. It misapplies probability theory, as the assumption that the odds have changed is wrong. In gambling, there really is no such thing as luck, although it might sometimes seem as if there is. But it is really all mathematics.