How to Calculate the Probability of an Event
The probability of an event tells you how likely that outcome is out of all the outcomes that could happen. Our probability calculator handles the arithmetic once you enter the number of favorable outcomes and total outcomes, but the underlying formula is simple enough to work out by hand for most everyday situations.
The basic formula
Probability of an event equals the number of favorable outcomes divided by the total number of possible outcomes. The result is always a number between 0 and 1, where 0 means the event can't happen and 1 means it's certain. Multiplying that result by 100 converts it into a percentage, which is often easier to talk about in everyday terms. Betting sites often quote the same idea as odds, which is a different number; odds vs probability shows how to convert between them.
A simple worked example
Consider rolling a standard six-sided die and wanting the probability of rolling a 4. There's exactly one favorable outcome, the number 4, out of six total possible outcomes. Dividing 1 by 6 gives about 0.167, or 16.7 percent. If instead you wanted the probability of rolling an even number, there are three favorable outcomes, 2, 4, and 6, out of six total, giving 3 divided by 6, or 0.5, which is 50 percent.
Independent events
Two events are independent when the outcome of one has no effect on the outcome of the other, such as flipping a coin twice. To find the probability that both independent events happen, multiply their individual probabilities together. The chance of flipping heads twice in a row is 0.5 times 0.5, which equals 0.25, or 25 percent. This multiplication rule extends to any number of independent events: rolling a 6 on a die three times in a row would be (1/6) times (1/6) times (1/6), or about 0.46 percent. If multiplying fractions feels rusty, how to multiply and divide fractions walks through it.
Dependent events
Dependent events are ones where an earlier outcome changes the probability of a later one, which commonly happens when drawing items without replacement. Picture a bag with 5 red marbles and 5 blue marbles. The probability of drawing a red marble first is 5 out of 10, or 0.5. If that marble isn't put back, the bag now has 4 red and 5 blue marbles left, so the probability of drawing red again is 4 out of 9. Multiplying 0.5 by 4/9 gives about 0.222, or 22.2 percent, for drawing two red marbles in a row without replacement.
The complement rule
Sometimes it's easier to calculate the probability that an event doesn't happen and subtract that from 1. This is called the complement rule: probability of an event happening equals 1 minus the probability of it not happening. If the probability of rain tomorrow is 30 percent, the probability of no rain is 70 percent, and the two always add up to 100 percent. This shortcut is especially useful for at least one questions, since calculating every possible way something could happen at least once is often more complicated than calculating the one way it doesn't happen at all.
Common mistakes to avoid
A frequent error is forgetting to adjust the total when events are dependent, which leads to treating a without-replacement scenario as if items were being replaced each time. Another common mistake is adding probabilities of independent events together instead of multiplying them, which only applies when finding the probability of one event or another happening, not both happening together. Confirming whether a situation involves combined events or alternative events, and whether earlier outcomes affect later ones, are the two questions worth checking before applying any formula.
Frequently asked questions
What's the difference between probability and odds?
Probability compares favorable outcomes to all possible outcomes, such as 1 out of 6 for rolling a specific number on a die. Odds compare favorable outcomes to unfavorable ones directly, which for the same die roll would be 1 to 5.
How do you find the probability of either of two events happening?
For mutually exclusive events, which can't happen at the same time, add their individual probabilities together. For events that can overlap, add the probabilities and then subtract the probability of both happening together, to avoid counting that overlap twice.
Can a probability be greater than 1 or negative?
No. A valid probability always falls between 0 and 1, or between 0 and 100 percent. A calculated result outside that range means there's an error in how the favorable or total outcomes were counted.