Conditional 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.
For conditional probability, the given event becomes the whole: divide by its row or column total. P(pet ∣ sport) = 820, but P(sport ∣ pet) = 812.
Formal definition and notation
For eventsA and B in a sample space with P(B) > 0, the conditional probability of A given B is P(A ∣ B) = P(A ∩ B)P(B). With equally likely outcomes this is the fraction of B‘s outcomes that are also in A: n(A ∩ B)n(B). Rearranged, it gives the multiplication rule P(A ∩ B) = P(B) · P(A ∣ B). Events are independent exactly when P(A ∣ B) = P(A).
Symbol
Read it as
Meaning
P(A ∣ B)
the probability of A given B
B is known to have happened; the vertical bar means “given”
P(A ∩ B)
the probability of A and B
both events happen
Key facts
The event after the bar is the condition: it becomes the new whole. Divide by its total, not by the grand total.
Order matters: P(A ∣ B) and P(B ∣ A) are usually different (0.4 and 0.67 in the table).
If P(A ∣ B) = P(A), then knowing B does not change the chance of A, and the events are independent.
Drawing without replacement creates conditional probabilities: after an ace is drawn, P(ace) = 351.
The multiplication rule P(A ∩ B) = P(B) · P(A ∣ B) is the same formula solved for the “and” probability.
probability of the given event; must be greater than 0
From a table use counts: P(pet ∣ sport) = 820 = 0.4.
Conditional Probability examples, step by step
Example 1: Roll a die
Warm-up
A fair die is rolled and lands on an even number. What is the probability that it is a 6?
Given “even”, the possible outcomes shrink to 2, 4, 6.
Only one of these is a 6.
P(6 ∣ even) = 13.
Answer: The probability is 13.
Example 2: Use the formula
Core
On a school day, the probability that it rains and the bus is late is 0.12. The probability that it rains is 0.30. Find the probability that the bus is late, given that it rains.
P(late ∣ rain) = P(late and rain)P(rain).
= 0.120.30.
= 0.4.
Answer: The probability is 0.4, or 40 percent.
Example 3: Without replacement
Test-style
A bag holds 5 red and 3 blue marbles. Two marbles are drawn without replacement. What is the probability that the second is red, given that the first was red? What is the probability that both are red?
After one red is removed, 4 red remain out of 7: P(R2 ∣ R1) = 47.
Multiplication rule: P(R1 and R2) = 58 × 47.
= 2056 = 514.
Answer:P(R2 ∣ R1) = 47 and P(both red) = 514.
Common mistakes
✗ MistakeP(pet ∣ sport) = 850
✓ InsteadDivide by the sport total: 820 = 0.4.
850 is P(pet and sport). “Given sport” means only the 20 sport players count.
✗ MistakeP(A ∣ B) = P(B ∣ A), so both table answers are 0.4.
✓ InsteadP(sport ∣ pet) = 812 ≈ 0.67, not 0.4.
The given event decides the denominator. Swapping the events changes it.
✗ MistakeP(A ∣ B) = P(A) × P(B)
✓ InsteadP(A ∣ B) = P(A ∩ B)P(B).
Multiplying probabilities gives an “and” probability (for independent events), not a conditional one.
For independent events the condition changes nothing: P(A ∣ B) = P(A). In the table, P(pet) = 0.24 but P(pet ∣ sport) = 0.4, so they are dependent.
Quick check: 3 questions
Question 1 of 3
If P(A and B) = 0.18 and P(B) = 0.6, what is P(A ∣ B)?
Show answer and explanation
Answer: A.0.180.6 = 0.3. 0.108 multiplies instead of dividing, 0.78 adds, and 3.33 divides the wrong way.
Question 2 of 3
Using the table of 50 students, what is the probability that a student plays a sport, given that the student has no pet?
Show answer and explanation
Answer: C. Restrict to the 38 students with no pet; 12 of them play a sport: 1238 = 619. 1250 uses the grand total, 1220 gives sport the role of the condition, and 2638 counts non-players.
Question 3 of 3
Events A and B have P(A) = 0.5 and P(A ∣ B) = 0.5. What can you conclude?
Show answer and explanation
Answer: D. Knowing B leaves the chance of A unchanged, which is the definition of independence. Mutually exclusive events would give P(A ∣ B) = 0, and nothing here fixes P(B).
The bar means “given”. P(A ∣ B) is the probability of A when you already know that B happened. It is not a fraction bar and not division. For example, P(rain ∣ cloudy) asks how likely rain is on days that are already cloudy, which is higher than the chance of rain on a random day.
How do you find conditional probability from a two-way table?
Find the row or column for the given condition and use its total as the denominator. Then take the count where both events happen as the numerator. In a table where 20 students play a sport and 8 of them have pets, P(pet ∣ sport) = 820 = 0.4. Ignore every row or column outside the condition.
What is the difference between conditional probability and independence?
Conditional probability is a number, P(A ∣ B). Independence is a relationship: A and B are independent when that number equals P(A), so learning B tells you nothing about A. Coin flips are independent; drawing cards without replacement is not, because the first card changes the deck.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSS-CP.A.3, CCSS HSS-CP.A.5, CCSS HSS-CP.B.6
Cite this entry
Effortless Math Team. “Conditional Probability.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/conditional-probability/
Flashcard
TermConditional ProbabilityP(A ∣ B)
DefinitionThe chance of A when B is known to have happened: P(A and B)P(B). Die shows even; chance it is 6: 13.
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, 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, 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, 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 4Independent 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 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 1Venn 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, StatisticsCircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.Geometry · Kindergarten, Grade 1, Grade 7Even 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 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 5Joint 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 1DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5NumeratorThe numerator is the top number in a fraction. It tells how many parts are being counted, while the denominator below the fraction bar tells how many equal parts make one whole. In 35, the numerator is 3: three of five equal parts are taken.Number System · Grade 3, Grade 4, Grade 5NumberA 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 2ProbabilityProbability 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 8Mutually 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, 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, Statistics