An 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.
Draw a loop around the outcomes that make the event happen. Its probability is the share of the sample space inside the loop.
Formal definition and notation
An event A is a subset of the sample space S of a chance process. Event Aoccurs on a trial when the outcome of that trial is in A. Its probability satisfies 0 ≤ P(A) ≤ 1; with equally likely outcomes, P(A) = n(A)n(S). The complement of A, written Ac or “not A,” contains the outcomes not in A, and P(not A) = 1 − P(A).
A compound event combines events with “and” or “or,” often across two or more steps, like rolling a 6 and flipping heads.
Eventvs.Event (everyday meaning)
In everyday speech an event is something planned, like a concert. In probability it is a set of possible results of a chance process.
Quick check: 3 questions
Question 1 of 3
You roll a number cube. Which set is the event “rolling a prime number”?
Show answer and explanation
Answer: B. The primes from 1 to 6 are 2, 3 and 5. 1 is not prime, {2, 4, 6} is the even numbers, and {3, 5} leaves out 2, the even prime.
Question 2 of 3
If P(A) = 0.35, what is P(not A)?
Show answer and explanation
Answer: C. An event and its complement add to 1: 1 − 0.35 = 0.65. 1.35 adds instead of subtracting, and 0.35 and 0.5 ignore the complement rule.
Question 3 of 3
Which event has probability 0 when you roll a number cube?
Show answer and explanation
Answer: D. An 8 is not in the sample space {1, 2, 3, 4, 5, 6}, so it is impossible. Rolling less than 7 is certain (probability 1), odd is 1/2 and a 6 is 1/6.
Event: frequently asked questions
What is a simple event?
A simple event contains exactly one outcome, such as rolling a 3 on a number cube, with probability 16. Events with more than one outcome, like rolling an odd number, are sometimes called compound in everyday teaching, though Common Core uses compound event mainly for combinations of events.
What is the complement of an event?
The complement of event A is everything in the sample space that is not in A. For rolling a number cube with A = “roll a 5 or 6,” the complement is {1, 2, 3, 4}. Since one of them must happen, P(not A) = 1 − P(A) = 1 − 13 = 23.
Sample 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, StatisticsProbabilityProbability 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.Probability · Grade 6, Grade 7, Grade 8SubsetA subset is a set whose elements all belong to another set. If every element of A is also in B, then A is a subset of B, written A ⊆ B. Every set is a subset of itself, the empty set is a subset of every set, and a set with n elements has 2n subsets.Probability · Algebra 1, StatisticsOutcomeAn 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, 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 2Compound EventA compound event is an event that combines two or more simple events, such as flipping heads and rolling a 6, or drawing a red card or a king. Find its probability by counting outcomes in a list, table or tree diagram, or with rules such as P(A and B) = P(A) × P(B) for independent events.Probability · Grade 7, Grade 8, 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, StatisticsWhole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5Odd NumberAn odd number is a whole number that cannot be split into two equal groups: dividing it by 2 always leaves a remainder of 1. Odd numbers end in 1, 3, 5, 7 or 9, such as 7, 23 and 409. Every odd number can be written as 2k + 1, one more than a multiple of 2.Number System · Grade 2, Grade 3, Grade 4DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5Even NumberAn even number is an integer that can be divided by 2 with no remainder, so it can be split into two equal groups or into pairs. Even numbers end in 0, 2, 4, 6 or 8, such as 14, 236 and −10. Zero is even. Every even number can be written as 2n for some integer n.Number System · Grade 2, Grade 3, Grade 4CombinationA combination is a selection of items from a group in which the order does not matter. Picking Ana, Ben and Cy for a team is the same combination as Cy, Ana and Ben. The number of ways to choose r items from n is C(n, r) = n!r!(n − r)!; for example, C(5, 2) = 10.Probability · Algebra 2, Precalculus, StatisticsIndependent 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 2Dependent 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, StatisticsVenn DiagramA Venn diagram is a picture that uses overlapping circles inside a rectangle to show how groups, or sets, are related. Each circle is a set, the overlap holds items in both sets, and the rectangle holds everything being considered. Venn diagrams make it easy to count “and”, “or” and “not” cases and to find probabilities.Probability · Grade 7, Algebra 2, Statistics