Probability in Math: Definition, Formula & Examples

Definitionnoun

What is probability in math?

Probability measures how likely an event is, from 0 for impossible to 1 for certain. For equally likely outcomes it is the number of favorable outcomes divided by the total number of outcomes: for a fair coin, the probability of heads is 12. Probabilities can be written as fractions, decimals or percents.

In Spanish: probabilidad
Also called: chance, likelihood
Grade level: 6-8

Grade 6Grade 7Grade 8CCSS 7.SP.C.5CCSS 7.SP.C.7.aGEDSATACT

Probability in a picture

Every probability sits between 0 (impossible) and 1 (certain). With 3 red marbles out of 4, red is likely, 34; the probabilities of all outcomes add to 1.

Formal definition and notation

For an experiment whose outcomes are all equally likely, the probability of an event A is P(A) = outcomes in Atotal outcomes. Every probability satisfies 0 ≤ P(A) ≤ 1, and the probabilities of all the outcomes in the sample space add to 1. The probability that A does not happen is 1 − P(A). When outcomes are not equally likely, probabilities come from a model, or are estimated from the relative frequency of the event in many trials (experimental probability).

Symbol	Read it as	Meaning
P(A)	P of A	the probability of event A, such as P(heads) = 12
P(not A)	P of not A	the probability that A does not happen: 1 − P(A)
0 ≤ P ≤ 1	P is between 0 and 1	the range of every probability (0% to 100%)

Key facts

Near 0 means unlikely, about 12 means neither likely nor unlikely, and near 1 means likely.
Complement rule: P(not A) = 1 − P(A). The chance of not rolling a 6 is 1 − 16 = 56.
The same probability can be written three ways: 14 = 0.25 = 25%.
Experimental probability is (times the event happened) ÷ (number of trials). With many trials it usually gets close to the theoretical probability.
For independent events, multiply: two fair coins both land heads with probability 12 × 12 = 14.

Theoretical probability

P(A) = favorable outcomestotal outcomes

favorable outcomes: the outcomes where event A happens
total outcomes: every possible outcome (the sample space), all equally likely

This works only when every outcome is equally likely. Rolling a 3 on a fair number cube: 1 favorable outcome out of 6, so P = 16.

Try it: spin and compare

Interactive

Spin once, 10, 100 or 1,000 times. The bars show how often each color came up; the dashed lines show the theoretical probabilities, 1/2, 1/3 and 1/6.

This interactive needs JavaScript. The worked examples below show the same ideas step by step.

Probability examples, step by step

Example 1: One fair number cube

Warm-up

A fair number cube is rolled. What is the probability of rolling a number greater than 4?

Total outcomes: 1, 2, 3, 4, 5, 6, which is 6 equally likely outcomes.
Favorable outcomes: 5 and 6, which is 2 outcomes.
P = 26 = 13.

Answer: The probability is 1/3, or about 0.33.

Example 2: Use the complement

Core

A bag has 5 red, 3 green and 2 yellow tiles. One tile is picked at random. What is the probability that it is not green?

Total tiles: 5 + 3 + 2 = 10.
P(green) = 310.
P(not green) = 1 − 310 = 710.
Check by counting: 5 red + 2 yellow = 7 tiles that are not green.

Answer: The probability of not green is 7/10, or 70%.

Example 3: Two number cubes

Test-style

Two fair number cubes are rolled. What is the probability that the sum is 7?

Each cube has 6 outcomes, so there are 6 × 6 = 36 equally likely pairs.
Pairs with sum 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1). That is 6 pairs.
P = 636 = 16.
Trap: there are 11 possible sums (2 to 12), but they are not equally likely, so the answer is not 111.

Answer: The probability of a sum of 7 is 1/6.

Common mistakes

✗ MistakeA bag has 2 red and 3 blue marbles, so P(red) = 23.

✓ InsteadP(red) = 22 + 3 = 25.

Divide by the total number of outcomes, not by the number of other outcomes. 23 is the ratio of red to blue.

✗ MistakeRain is very likely tomorrow, so its probability is 1.5.

✓ InsteadA very likely event has a probability close to 1, such as 0.9 (90%).

A probability cannot be less than 0 or greater than 1.

✗ MistakeA fair coin landed heads 5 times in a row, so tails is now more likely.

✓ InsteadThe coin has no memory: the next flip is still 12 heads and 12 tails.

Independent trials do not affect each other. Past results do not change the next probability.

✗ MistakeA spinner is half red, a quarter blue and a quarter green, so P(red) = 13 because there are 3 colors.

✓ InsteadP(red) = 12: red covers half the spinner.

Favorable over total works only for equally likely outcomes. Three colors are not equally likely when the sections differ in size.

Not to be confused with

Probabilityvs.Odds

Odds compare favorable to unfavorable outcomes; probability compares favorable to all outcomes. With 2 red and 3 blue marbles, the odds of red are 2 to 3, but the probability is 25.

Probabilityvs.Experimental probability

Theoretical probability comes from reasoning (16 for a 3 on a die). Experimental probability comes from trials: 9 threes in 60 rolls gives 960 = 0.15.

Probabilityvs.Ratio

A probability is a special part-to-whole ratio that always lies between 0 and 1. A ratio can also compare part to part, like 2 : 3, and can be any size.

Quick check: 3 questions

Question 1 of 3

A spinner has 8 equal sections numbered 1 to 8. What is the probability of landing on a multiple of 3?

Show answer and explanation

Answer: A. The multiples of 3 from 1 to 8 are 3 and 6, so P = 28 = 14. Choice B also counts 9, which is not on the spinner; C counts only the 3; and D divides the favorable outcomes by the unfavorable ones.

Question 2 of 3

Which number cannot be a probability?

Show answer and explanation

Answer: C. 54 = 1.25, which is greater than 1, so it cannot be a probability. 0 is an impossible event, 0.99 is almost certain, and 45% = 0.45 is in range.

Question 3 of 3

The probability of rain tomorrow is 0.35. What is the probability that it does not rain?

Show answer and explanation

Answer: B. Use the complement: 1 − 0.35 = 0.65. Choice A repeats the chance of rain, C adds instead of subtracting (and is above 1), and D says no rain is impossible.

Probability: frequently asked questions

What is the difference between probability and odds?

Probability compares favorable outcomes with all outcomes; odds compare favorable outcomes with unfavorable ones. With 1 winning ticket out of 5, the probability of winning is 15, but the odds are 1 to 4. To turn odds of a to b into a probability, use aa + b.

Can a probability be written as a percent?

Yes. Multiply the fraction or decimal by 100: 34 = 0.75 = 75%. Weather forecasts do this when they say a 30% chance of rain, which is a probability of 0.3. Whatever the form, the value must stay between 0 (0%) and 1 (100%).

What is the difference between theoretical and experimental probability?

Theoretical probability is what should happen, worked out by counting equally likely outcomes: 12 for heads. Experimental probability is what did happen in trials: 27 heads in 50 flips is 2750 = 0.54. With more trials, the experimental value usually moves closer to the theoretical one.

Practice and tools

Probability practicePractice on Effortless Math Probability CalculatorSingle, complement and combined events Dice RollerRun your own experiments and compare with theory

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.SP.C.5, CCSS 7.SP.C.7.a

Cite this entry

Effortless Math Team. “Probability.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/probability/

Flashcard

TermProbabilityP(A), from 0 to 1

DefinitionHow likely an event is: favorable outcomes ÷ total outcomes. Rolling a 3 on a die: 16.

Study the full deck

Browse all terms in the Math Dictionary · More probability terms · Random term

OutcomeAn outcome is one possible result of a chance experiment, such as rolling a 4 on a number cube or getting tails on a coin flip. The set of all outcomes is the sample space, and a group of outcomes is an event. When outcomes are equally likely, each one has probability 1 divided by the number of outcomes.Probability · Grade 7, Grade 8, StatisticsEventAn event is a set of one or more outcomes of a chance experiment, such as “rolling an even number” on a number cube. It is part of the sample space. When outcomes are equally likely, the probability of an event is the number of outcomes in it divided by the number of outcomes in the sample space.Probability · Grade 7, Grade 8, StatisticsSample SpaceThe sample space of a chance experiment is the set of all its possible outcomes, usually written S. For one coin flip, S = {H, T}; for one number cube, S = {1, 2, 3, 4, 5, 6}. Organized lists, tables and tree diagrams help you write it out, and the probabilities of all its outcomes add to 1.Probability · Grade 7, Grade 8, StatisticsRelative FrequencyRelative frequency is how often a value or event occurs compared with the total number of observations: frequency ÷ total. It can be a fraction, a decimal or a percent. If 12 of 40 students choose apples, the relative frequency of apples is 1240 = 0.3, or 30%. All the relative frequencies in a table add to 1.Statistics · Grade 7, Grade 8, StatisticsNumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Independent EventsIndependent events are two events where the outcome of one does not change the probability of the other. Flipping heads on a coin does not affect rolling a 5 on a die. For independent events, multiply: P(A and B) = P(A) × P(B), so P(heads and 5) = 12 × 16 = 112.Probability · Grade 7, Algebra 1, Algebra 2RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8OddsOdds compare the number of ways an event can happen with the number of ways it cannot, written as a ratio such as 3 to 2 or 3:2. Odds in favor are favorable : unfavorable, and odds against are the reverse. Probability instead divides favorable outcomes by all outcomes, so 3:2 odds mean a probability of 35.Probability · Grade 7, StatisticsPercentA percent is a number out of 100. Percent means per hundred, so 25% means 25 out of 100, or 25100 = 0.25. To find a percent of a quantity, change the percent to a decimal and multiply: 20% of 80 is 0.20 × 80 = 16. The symbol for percent is %.Number System · Grade 5, Grade 6, Grade 7DecimalA decimal is a number written with a decimal point to show parts of a whole. Decimals use place value to represent tenths, hundredths, thousandths and more: in 3.14, the 1 means 1 tenth and the 4 means 4 hundredths. So 0.75 = 75100, and 0.5 is one half.Number System · Grade 4, Grade 5, Grade 6MeanThe mean is the sum of the values divided by the number of values. It is the arithmetic average: add every value and divide by how many there are. For 3, 7, 5 and 9, the mean is 24 ÷ 4 = 6. The mean is the balance point of the data, so one outlier can pull it far.Statistics · Grade 5, Grade 6, Grade 7Dependent EventsDependent events are two events where the outcome of the first changes the probability of the second. Drawing two cards without putting the first one back is the classic example. For dependent events, P(A and B) = P(A) × P(B ∣ A), where P(B ∣ A) is the probability of B once A has happened.Probability · Grade 7, Algebra 2, StatisticsMutually Exclusive EventsMutually exclusive events are events that cannot happen at the same time, so P(A and B) = 0. Rolling a 2 and rolling a 5 on one roll of a number cube are mutually exclusive. For such events the addition rule is simple: P(A or B) = P(A) + P(B).Probability · Grade 7, Algebra 2, Statistics

Probability

What is probability in math?

Probability in a picture

Formal definition and notation

Key facts

Theoretical probability

Try it: spin and compare

Probability examples, step by step

Example 1: One fair number cube

Example 2: Use the complement

Example 3: Two number cubes

Common mistakes

Not to be confused with

Quick check: 3 questions

Probability: frequently asked questions

What is the difference between probability and odds?

Can a probability be written as a percent?

What is the difference between theoretical and experimental probability?

Related terms

Practice and tools

Sources

Flashcard

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