Independent Events in Math: Definition, Formula & Examples

Definitionnoun phrase

What are independent events in probability?

Independent 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.

In Spanish: eventos independientes
Also called: independent event, independence, independent probability
Grade level: Statistics

Grade 7Algebra 1Algebra 2StatisticsCCSS HSS-CP.A.2SATACTGEDASVABHiSETAP

Independent Events in a picture

Heads-and-5 is 1 of 12 equally likely outcomes, exactly 12 × 16. Putting a marble back keeps draws independent; keeping it out makes them dependent.

Formal definition and notation

Events A and B are independent if P(A ∩ B) = P(A) · P(B). When P(B) > 0, this is the same as P(A ∣ B) = P(A): knowing that B happened leaves the probability of A unchanged. Events that are not independent are dependent.

Symbol	Read it as	Meaning
P(A and B)	the probability of A and B	both events happen; also written P(A ∩ B)
P(A ∣ B)	the probability of A given B	the probability of A once you know B happened
P(A ∩ B) = P(A)P(B)	P of A and B equals P of A times P of B	the test for independence

Key facts

Ask: does knowing the first result change the chances for the second? If not, the events are independent.
Separate coins, dice and spinners are independent. So are draws with replacement, because the bag is reset each time.
Draws without replacement are dependent: P(red, red) from 3 red and 5 blue is 38 × 27, not 38 × 38.
Past results do not change independent future ones: after five heads in a row, the next flip is still 12 heads.
Independent is not the same as mutually exclusive. Two events that cannot both happen (with nonzero probabilities) are always dependent.
To test data, compare P(A and B) with P(A) × P(B), for example from a two-way table.

Multiplication rule for independent events

P(A and B) = P(A) × P(B)

P(A): the probability of the first event
P(B): the probability of the second event
P(A and B): the probability that both happen

It extends to more events: three independent coin flips all landing heads is 12 × 12 × 12 = 18. Use it only when the events really are independent.

Independent Events examples, step by step

Example 1: Coin and spinner

Warm-up

A spinner has 4 equal sections, one of them blue. You flip a coin and spin. What is P(tails and blue)?

The coin and the spinner do not affect each other, so the events are independent.
P(tails) = 12 and P(blue) = 14.
Multiply: 12 × 14 = 18.

Answer: P(tails and blue) = 18, or 0.125.

Example 2: With and without replacement

Core

A bag holds 4 green and 6 yellow tiles. You draw two tiles. Find P(green, then green) (a) if you replace the first tile and (b) if you do not.

(a) With replacement the bag is the same each time: 410 × 410 = 16100 = 425.
(b) Without replacement, after one green there are 3 green out of 9: 410 × 39 = 1290 = 215.
Only (a) uses independent events; in (b) the first draw changes the second.

Answer: (a) 425 = 0.16; (b) 215 ≈ 0.133.

Example 3: Test for independence

Test-style

In a school, 40% of students play a sport, 50% are in band, and 20% do both. Are playing a sport and being in band independent?

P(S) = 0.4 and P(B) = 0.5, so P(S) × P(B) = 0.2.
P(S and B) = 0.2.
The two numbers are equal, so the events are independent.
Check another way: P(S ∣ B) = 0.20.5 = 0.4, the same as P(S).

Answer: Yes, they are independent, because 0.4 × 0.5 = 0.2 equals P(both).

Common mistakes

✗ MistakeTwo cards drawn without replacement from a deck: P(ace, ace) = 452 × 452.

✓ Instead452 × 351 = 1221.

Without replacement the second draw has one fewer card and one fewer ace. The events are dependent, so the second probability must change.

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

✓ InsteadP(tails) = 12 on every flip.

Coin flips are independent. The coin has no memory; this mistake is called the gambler’s fallacy.

✗ MistakeFor independent events, P(A and B) = P(A) + P(B).

✓ InsteadMultiply: P(A and B) = P(A) × P(B).

“And” with independent events multiplies. Adding is for “or” with events that cannot happen together, and it can give totals above 1.

Not to be confused with

Independent Eventsvs.Mutually exclusive events

Mutually exclusive events cannot happen together, so P(A and B) = 0. Rolling a 2 and rolling a 5 on one die are mutually exclusive, and therefore dependent: if one happens, the other cannot.

Independent Eventsvs.Dependent events

Dependent events affect each other’s probabilities, like two draws without replacement. Then P(A and B) = P(A) × P(B ∣ A).

Independent Eventsvs.Probability

Probability is the chance of one event; independence describes how two events relate to each other.

Quick check: 3 questions

Question 1 of 3

Which pair of events is independent?

Show answer and explanation

Answer: B. The first roll does not change the second, so two die rolls are independent. The card draws and the student picks remove an option, and rain directly makes the ground wet, so those pairs are dependent.

Question 2 of 3

P(A) = 0.3 and P(B) = 0.6. If A and B are independent, what is P(A and B)?

Show answer and explanation

Answer: B. Multiply: 0.3 × 0.6 = 0.18. 0.9 adds the probabilities, 0.3 is just P(A), and 0.5 is the ratio 0.30.6.

Question 3 of 3

You roll a fair die three times. What is the probability of getting a 6 every time?

Show answer and explanation

Answer: D. The rolls are independent: 16 × 16 × 16 = 1216. 36 adds the probabilities and 118 multiplies 6 by 3 instead of cubing it.

Independent Events: frequently asked questions

How do you know if two events are independent?

Ask whether one outcome changes the chances of the other. If you can compute probabilities, check whether P(A and B) = P(A) × P(B); if the two sides are equal, the events are independent. Another test is P(A ∣ B) = P(A): learning that B happened does not change the probability of A.

What is the difference between independent and mutually exclusive events?

Independent events can happen together and do not affect each other, like a coin flip and a die roll. Mutually exclusive events cannot happen together, like rolling a 1 and a 6 on the same roll. If two events with nonzero probabilities are mutually exclusive, they are dependent, because one happening makes the other impossible.

Is drawing with replacement independent?

Yes. Putting the item back restores the original set, so the second draw has exactly the same probabilities as the first. Without replacement, the set shrinks and its makeup changes, so the second probability depends on what you drew first. That is why without-replacement problems use 38 × 27 instead of 38 × 38.

Practice and tools

Probability practicePractice on Effortless Math Probability CalculatorProbabilities of A and B, A or B, for independent events Dice RollerRoll dice many times and watch independence in action

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSS-CP.A.2

Cite this entry

Effortless Math Team. “Independent Events.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/independent-events/

Flashcard

TermIndependent EventsP(A)P(B)

DefinitionEvents where one outcome does not change the other’s probability. Multiply: coin heads and die 5 is 12 × 16 = 112.

Study the full deck

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

ProbabilityProbability 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 8EventAn 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, StatisticsDataData are facts or values collected to answer a question, such as survey answers, measurements or counts. Numerical data are numbers you measure or count, like heights; categorical data are labels or groups, like favorite colors. Data are organized in tables and shown in graphs so patterns are easy to see.Statistics · Grade 2, Grade 3, Grade 6Two-Way TableA two-way table shows counts for two categorical variables collected from the same group, with one variable in the rows and the other in the columns. Each inside cell is a joint frequency, and the totals in the margins are marginal frequencies. Dividing a cell by a row or column total gives relative frequencies that reveal associations.Statistics · Grade 8, Algebra 1NumberA 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 2Mutually 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, StatisticsDependent 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, 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, StatisticsSideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2FractionA fraction is a number that names equal parts of a whole, written ab. The bottom number, the denominator, tells how many equal parts make one whole; the top number, the numerator, tells how many of those parts you have. So 34 means 3 of 4 equal parts. Every fraction is also a point on the number line.Number System · Grade 3, Grade 4, Grade 5Conditional ProbabilityConditional probability is the probability that an event happens given that another event has already happened. It is written P(A ∣ B), read “the probability of A given B”, and found by P(A ∣ B) = P(A and B)P(B). Knowing B shrinks the possible outcomes to only those in B.Probability · Algebra 2, StatisticsCompound 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, StatisticsJoint FrequencyJoint frequency is the number of data values that fit two categories at the same time, such as students who are in Grade 7 and play an instrument. In a two-way table, a joint frequency is one of the inside cells, not the totals. Dividing it by the grand total gives the joint relative frequency.Statistics · Grade 8, Algebra 1

Independent Events

What are independent events in probability?

Independent Events in a picture

Formal definition and notation

Key facts

Multiplication rule for independent events

Independent Events examples, step by step

Example 1: Coin and spinner

Example 2: With and without replacement

Example 3: Test for independence

Common mistakes

Not to be confused with

Quick check: 3 questions

Independent Events: frequently asked questions

How do you know if two events are independent?

What is the difference between independent and mutually exclusive events?

Is drawing with replacement independent?

Related terms

Practice and tools

Sources

Flashcard

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