Dependent 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.
In Spanish
eventos dependientes
Also called
dependent probability, events that are not independent
The second-draw probabilities change depending on the first draw: 2/4 after a red but 3/4 after a blue. That change is what makes the events dependent. Multiply along a path to get its probability.
Formal definition and notation
EventsA and B are dependent when knowing that one occurred changes the probability of the other: P(B ∣ A) ≠ P(B). Equivalently, P(A and B) ≠ P(A) · P(B). Here P(B ∣ A) = P(A and B)P(A), for P(A) > 0, is the conditional probability of B given A. Rearranged, it gives the general multiplication rule, P(A and B) = P(A) · P(B ∣ A), which works for any two events, dependent or not.
Symbol
Read it as
Meaning
P(B ∣ A)
the probability of B given A
the probability of B once you know A has happened
P(A and B)
the probability of A and B
the probability that both happen; also written P(A ∩ B)
Key facts
Drawing without replacement makes the draws dependent: the first draw changes what is left. Drawing with replacement keeps them independent.
To test two events, compare P(B ∣ A) with P(B). If they differ, the events are dependent.
After each draw, both the number of favorable items and the total go down when the first draw was favorable; only the total goes down when it was not.
Tree diagrams and two-way tables are the easiest ways to see dependent probabilities.
Mutually exclusive events that each have a nonzero probability are always dependent: if one happens, the other has probability 0.
Multiplication rule for dependent events
Multiplication rule for dependent events
P(A and B) = P(A) × P(B ∣ A)
P(A)
the probability of the first event
P(B ∣ A)
the probability of the second event, recomputed after the first has happened
For three draws, keep chaining: P(A and B and C) = P(A) · P(B ∣ A) · P(C ∣ A and B). Each new factor uses what is left.
Dependent Events examples, step by step
Example 1: Two marbles without replacement
Warm-up
A bag holds 4 green and 6 yellow marbles. You draw 2 marbles without replacing the first. What is the probability that both are green?
First draw: P(G) = 410.
One green is gone, so 3 greens are left out of 9 marbles: P(G ∣ G) = 39.
Multiply: 410 × 39 = 1290 = 215.
Answer:P(both green) = 215, about 0.13.
Example 2: Two aces from a deck
Core
Two cards are drawn from a standard 52-card deck without replacement. What is the probability that both are aces?
First card: 4 aces out of 52, so 452.
Second card: 3 aces are left out of 51, so 351.
Multiply: 452 × 351 = 122652 = 1221.
Answer:P(two aces) = 1221, about 0.0045.
Example 3: Test for dependence with data
Test-style
In a class of 30 students, 18 play a sport and 14 play an instrument. Of the 18 who play a sport, 12 also play an instrument. Are the events plays a sport and plays an instrument independent or dependent?
Probability of playing an instrument for the whole class: P(I) = 1430 = 715 ≈ 0.47.
Probability among the athletes: P(I ∣ S) = 1218 = 23 ≈ 0.67.
P(I ∣ S) ≠ P(I): knowing a student plays a sport changes the chance they play an instrument.
Answer: The events are dependent, because P(I ∣ S) = 23 but P(I) = 715.
Common mistakes
✗ MistakeFor two greens without replacement from 4 green and 6 yellow: 410 × 410 = 425.
✓ Instead410 × 39 = 215.
Without replacement, the first green is gone. The second fraction must use what is left: 3 greens out of 9 marbles.
✗ MistakeThinking a coin is more likely to land tails after five heads in a row.
✓ InsteadEach flip is still 1/2 heads, 1/2 tails.
Coin flips are independent. The coin has no memory, so earlier flips cannot change the next one.
✗ MistakeMixing up P(B ∣ A) and P(A ∣ B): the share of athletes who play an instrument, 1218, and the share of musicians who play a sport, 1214.
✓ InsteadP(I ∣ S) = 1218 and P(S ∣ I) = 1214 are different numbers.
The event after the bar is the one you already know happened; it sets the new total in the denominator.
Independent events do not affect each other: rolling a 6 does not change the next roll. Then P(A and B) = P(A) × P(B). Dependent events need P(B ∣ A) in place of P(B).
Mutually exclusive events cannot happen together, so P(A and B) = 0, like rolling a 2 and a 5 on one roll. Dependent events can happen together; one just changes the other’s chance.
Dependent Eventsvs.Conditional probability
Conditional probability, P(B ∣ A), is a number: the chance of B given A. Dependent describes a relationship: the events are dependent when that number differs from P(B).
Quick check: 3 questions
Question 1 of 3
Which pair of events is dependent?
Show answer and explanation
Answer: C. Keeping the first card changes what is left in the deck, so the second draw depends on the first. In the other three, nothing about the first action changes the chances for the second.
Question 2 of 3
A box has 5 red chips and 3 white chips. Two chips are drawn without replacement. What is the probability of red first, then white?
Show answer and explanation
Answer: A.58 × 37 = 1556. 1564 uses 38 for the second draw, as if the red chip were put back; 27 and 815 do not come from multiplying the two draws.
Question 3 of 3
P(A) = 0.5 and P(B ∣ A) = 0.3. What is P(A and B)?
Show answer and explanation
Answer: D. Multiplication rule: 0.5 × 0.3 = 0.15. 0.8 adds the probabilities, 0.2 subtracts them, and 0.6 divides 0.3 by 0.5.
Dependent Events: frequently asked questions
How do you know if events are dependent or independent?
Ask whether the first event changes the situation for the second. Drawing without replacement, choosing people one at a time, or using survey data where groups differ usually signals dependence. To check with numbers, compare P(B ∣ A) with P(B), or P(A and B) with P(A) × P(B). Equal means independent; different means dependent.
What is the formula for dependent events?
P(A and B) = P(A) × P(B ∣ A). Find the probability of the first event, then find the probability of the second event in the changed situation, and multiply. For example, two kings from a deck without replacement: 452 × 351 = 1221.
Can dependent events be mutually exclusive?
Yes. In fact, two mutually exclusive events that each have a nonzero probability are always dependent. If A happens, B cannot, so P(B ∣ A) = 0, which is different from P(B). Rolling a 1 and rolling a 6 on the same roll are mutually exclusive and dependent.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.SP.C.8, CCSS HSS-CP.A.2, CCSS HSS-CP.A.3, CCSS HSS-CP.B.8
Cite this entry
Effortless Math Team. “Dependent Events.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/dependent-events/
Flashcard
TermDependent Eventswithout replacement
DefinitionEvents where the first changes the chance of the second. Two reds from 3 red and 2 blue, no replacing: 35 × 24 = 310.
EventAn 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, 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 8Conditional 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, StatisticsMultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4NumberA 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 2Two-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 1Mutually 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, StatisticsDenominatorThe 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 5Independent 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 2DataData 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 6RatioA 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 8Sample 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, 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 1CombinationA 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, Statistics