Conditional Probability
Probability of A given B has occurred.
Formula
Variables
Example
Drawing 2 aces from a deck: P(2nd ace | 1st ace) = 3/51 ≈ 0.059.
How it works
Conditional probability tells you how likely event A is after you already know event B has happened. It works by restricting attention to the outcomes inside B, so the chance of A is measured as the fraction of B-outcomes that also satisfy A. The denominator is P(B) because you are now working within the smaller sample space where B is true.
When to use it
Use this when one event changes the sample space for another, such as medical test results, card draws without replacement, weather given a forecast condition, or any situation where you know one event has already occurred. It is especially useful for updating probabilities when new information is available.
Worked example, step by step
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. We want P(blue | not red). Step 1: Find P(blue and not red). Since blue marbles are not red, this is just P(blue) = 3/10. Step 2: Find P(not red) = (3 blue + 2 green)/10 = 5/10. Step 3: Apply the formula: P(blue | not red) = P(blue ∩ not red) / P(not red) = (3/10) / (5/10) = 3/5. Final answer: P(blue | not red) = 3/5 = 0.6.
Common mistakes to avoid
- Using P(A) instead of P(A ∩ B) in the numerator
- Forgetting that the denominator becomes P(B), not the probability of the whole original sample space
- Treating conditional probability as symmetric, so P(A|B) is assumed equal to P(B|A)
Did You Know?
Conditional probability is why "rare disease, positive test" often still means you're probably fine.
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More in Statistics & Probability
View allArithmetic Mean
BasicThe average of a data set.
Median
BasicThe middle value separating the higher and lower halves.
Variance
IntermediateAverage squared deviation from the mean.
Standard Deviation
IntermediateSquare root of the variance; typical spread.
Z-Score
IntermediateNumber of standard deviations a value is from the mean.
Bayes' Theorem
IntermediateUpdates a probability given new evidence.