Probability

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Flip a coin. Will it land on heads or tails? You cannot know for sure before it lands, but you do know something useful: heads and tails are equally likely. Probability is the part of math that turns feelings like "equally likely" or "pretty unlikely" into an exact number, so you can compare different chances fairly.

Turning "maybe" into a number

Probability is usually written as a number between 0 and 1, or as a percent between 0 percent and 100 percent.

• A probability of 0 means something is impossible. It will never happen.
• A probability of 1, or 100 percent, means something is certain. It will always happen.
• A probability of one half, or 50 percent, means it is equally likely to happen or not, like a fair coin landing on heads.

To find the probability of something simple, you count the ways it can happen and compare that to every way things could possibly turn out. A coin has 2 sides, so heads is 1 out of those 2 possible outcomes, giving a probability of one half. A six sided die has 6 faces, so rolling exactly a 4 is 1 out of 6, or about 17 percent.

Chances add up over many tries

A single coin flip cannot prove much. If you flip a coin 4 times, getting 3 heads would not be shocking. But if you flip that same coin 1,000 times, you would expect somewhere close to 500 heads, since each flip is equally likely to be heads or tails. The more times you repeat something random, the closer the results tend to drift toward what the probability predicts. This pattern is such a reliable rule that mathematicians gave it a name, the law of large numbers, and it is also the entire idea behind a Monte Carlo Simulation, which answers hard questions by trying something random an enormous number of times and looking at the results.

Independent events

Two events are called "independent" when one of them does not affect the other at all. A coin has no memory, so flipping tails does not make heads any more or less likely on the very next flip. Each flip's probability stays exactly one half, no matter what happened before. That is different from a Markov chain, where the very next step depends on the situation you are in right now, rather than being totally unaffected by anything.

Where probability shows up

Probability quietly shapes a lot of everyday life:

• A weather forecast saying "70 percent chance of rain" is a probability, built from studying many past days that started out looking like today.
• Insurance companies figure out how likely a car crash or a house fire is, and use that probability to decide what to charge for insurance.
• Doctors and scientists use probability to judge whether a new medicine is truly working or whether the results could just be luck.
• Card games and board games are full of probability, from the chance of drawing a certain card to the odds of rolling doubles.

Fun facts

• Much of the math of probability was born from a gambling question. In 1654, a gambler asked the mathematicians Blaise Pascal and Pierre de Fermat how to fairly split the pot in an unfinished game of chance, and their letters back and forth laid the foundation for the entire subject.
• Winning a typical big lottery jackpot is so unlikely that you are far more likely to be struck by lightning at some point in your life than to win it.
• Casinos are built entirely around probability. The games are designed so that, over many, many plays, the casino is mathematically likely to come out ahead, even though any single player can still get lucky.