Chance as a number
Probability is a way of putting a number on how likely something is. The scale runs from 0 to 1:
- 0 means the event cannot happen.
- 1 means the event is certain.
- Numbers in between describe how likely it is, with larger numbers meaning more likely.
You can write a probability as a fraction, a decimal or a percentage. A probability of 1/2 is the same as 0.5 and 50%. If a weather forecast says there is a 30% chance of rain, that is a probability of 0.3. It does not guarantee that it will rain on any particular day, only that in situations like this one, rain tends to occur on about three out of ten.
Probability describes uncertainty. It does not predict what will happen on one occasion, and it does not tell you how to beat the odds in a game of chance. Outcomes remain uncertain, which is the whole point.
Some vocabulary
- An experiment or trial is any process with an uncertain result, such as flipping a coin or rolling a die.
- An outcome is one possible result, such as “heads”.
- An event is a set of outcomes you care about, such as “an even number on a die”.
- The sample space is the list of all possible outcomes.
The basic formula
When all outcomes are equally likely, the probability of an event is:
probability = (number of outcomes in the event) ÷ (total number of outcomes)
Coin example
A fair coin has two outcomes, heads and tails. The probability of heads is 1 ÷ 2 = 0.5.
Die example
A fair six-sided die has outcomes 1, 2, 3, 4, 5 and 6.
- The probability of rolling a 4 is 1/6, which is about 0.167.
- The probability of an even number (2, 4 or 6) is 3/6 = 1/2.
- The probability of a number greater than 4 (5 or 6) is 2/6 = 1/3.
- The probability of rolling a 7 is 0/6 = 0, since it cannot happen.
- The probability of rolling a number from 1 to 6 is 6/6 = 1.
The probabilities of all the individual outcomes add up to 1. For the die, six outcomes of 1/6 each give 6/6 = 1. This is a useful check.
The complement: “not happening”
The probability that an event does not happen is 1 minus the probability that it does. If the chance of rolling a six is 1/6, then the chance of not rolling a six is 1 − 1/6 = 5/6. This shortcut is helpful when counting the outcomes you want would take longer than counting the ones you do not.
Combining events
“Or” for events that cannot both happen
For a single roll, “roll a 1 or a 2” has probability 1/6 + 1/6 = 2/6 = 1/3. You add the probabilities when the events cannot occur together.
“And” for independent events
Two events are independent if one does not affect the other. Flipping a coin twice is an example. The probability of two heads in a row is 1/2 × 1/2 = 1/4. You multiply the probabilities.
Another case: rolling a six on one die and a six on another die has probability 1/6 × 1/6 = 1/36.
To see this by counting, list the outcomes for two dice. There are 6 × 6 = 36 equally likely pairs. Exactly one is (6, 6), so the probability is 1/36. A total of 7 can be made from (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1), which is 6 pairs, so the probability is 6/36 = 1/6. A total of 12 comes from only (6,6), so it is 1/36. That is why 7 shows up more often than 12 when two dice are rolled repeatedly.
Theoretical and experimental probability
The calculations above give theoretical probability, based on reasoning about equally likely outcomes. Experimental probability is based on actual trials: the number of times an event happened divided by the number of trials.
If you flip a coin 10 times, you might get 7 heads. That does not mean the coin is unfair. Small samples vary a lot. With many more flips, the proportion of heads usually settles closer to one half. This idea, that results tend to approach the theoretical value over many trials, is sometimes called the law of large numbers. It describes a tendency over the long run and does not guarantee any particular result.
Running your own experiment is a good way to feel this. Flip a coin 20 times and record the results, then pool results with a friend and see how the proportion changes. The idea of designing and carrying out a test properly is covered under controlled experiments.
Common misunderstandings
The gambler’s fallacy. After five heads in a row, many people feel that tails is “due”. For a fair coin, each flip is independent, so the next flip is still 50% heads. The coin has no memory. The belief that it does is one of the logical fallacies people fall into most often.
Ignoring how the situation is set up. The coin and dice examples assume fair, equally likely outcomes. Many real situations do not have equal chances, and their probabilities have to be estimated from data rather than counted.
Probability in everyday decisions
Probability helps in reading forecasts, understanding test results, reading survey claims and weighing risks. It is worth being careful: a probability describes a group of similar cases, not a promise about one case. Games of chance are designed so that outcomes remain uncertain, and nothing in basic probability offers a method to win reliably.
Probabilities are also connected to fractions and percentages. Converting between them, covered in the article on fraction conversion, lets you describe 1/8 as 0.125 or 12.5%.