Conditional Probability Calculator

Calculate P(A|B) - the probability of A given B - from probabilities or frequency counts.

P(A∩B) — Joint Probability (%)

P(B) — Marginal Probability of B (%)

P| What is Conditional Probability?

Conditional probability P(A|B) is the probability that event A occurs given that event B has already occurred. The vertical bar “|” is read as “given.” It restricts the sample space to only those outcomes where B is true, then measures the fraction of those where A is also true. The formula is: P(A|B) = P(A∩B) ÷ P(B), where P(A∩B) is the joint probability (both A and B occur) and P(B) is the marginal probability (B occurs regardless of A).

A critical warning: P(A|B) ≠ P(B|A) in general. This confusion — called the base rate fallacy or transposing the conditional — is extremely common. “The probability of a fire alarm ringing given there is a fire” is not the same as “the probability of a fire given the alarm is ringing.” A very sensitive alarm might ring for any smoke, so P(alarm | fire) = 95% but P(fire | alarm) might be only 3% in a building with many false alarms. Bayes’ theorem provides the rigorous way to go from one to the other.

Conditional probability is the foundation of Bayesian reasoning — the framework for updating beliefs in light of new evidence. Starting from a prior probability P(A), observing evidence B, and computing a posterior P(A|B) is the essence of rational belief update. This process appears everywhere: medical diagnosis (what is P(disease | symptoms)?), spam filtering (P(spam | keywords)), weather forecasting (P(rain | pressure, humidity)), and machine learning classifiers.

The 2×2 contingency table is the most natural way to compute conditional probability from real data. Rows represent one event (A, ¬A), columns represent another (B, ¬B), and cells hold observed counts. From the table, P(A|B) = count(A∩B) ÷ count(B) — the proportion of “B column” observations that also have A. This calculator supports both the formula-based approach (when you know the probabilities directly) and the frequency-based approach (when you have raw data).

📐 Formulas

P(A|B) = P(A∩B) ÷ P(B)

P(A∩B) = joint probability (both A and B occur)

P(B) = marginal probability of B (P(B) > 0)

Check: P(A|B) + P(¬A|B) = 1 (100%)

Example: P(rain∩cloudy) = 0.30, P(cloudy) = 0.40 → P(rain|cloudy) = 0.30/0.40 = 75%

Bayes’ Theorem: P(A|B) = P(B|A) × P(A) ÷ P(B)

P(A) = prior probability · P(A|B) = posterior probability

P(B) = P(B|A)×P(A) + P(B|¬A)×P(¬A) [law of total probability]

Example: Disease prevalence 1%, test sensitivity 99%, specificity 95%. P(disease|positive) = (0.99×0.01) / (0.99×0.01 + 0.05×0.99) ≈ 16.7%

📖 How to Use This Calculator

Direct Input Mode

Enter P(A∩B) as a percentage — the probability both events occur (must be ≤ P(B)).

Enter P(B) as a percentage — the probability of the conditioning event (must be > 0).

Click Calculate to see P(A|B), P(¬A|B), joint probability, and the verification that P(A|B) + P(¬A|B) = 100%.

Contingency Table Mode

Enter the four cell counts: (A∩B), (¬A∩B), (A∩¬B), (¬A∩¬B) from your frequency data.

Click Calculate to see P(A|B) and P(B|A) - both conditional directions - plus the joint and marginal probabilities from the table.

💡 Example Calculations

Example 1 — Weather: Rain Given Clouds

P(rain ∩ cloudy) = 30%, P(cloudy) = 40%

P(rain | cloudy) = 0.30 / 0.40 = 0.75

P(¬rain | cloudy) = 1 − 0.75 = 0.25. Check: 75% + 25% = 100% ✓

P(rain | cloudy) = 75%. On cloudy days, it rains 75% of the time.

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Example 2 — Medical Test (Contingency Table)

1,000 patients: 50 sick & positive, 10 sick & negative, 150 healthy & positive, 790 healthy & negative

P(sick | positive) = 50 / (50 + 150) = 50/200 = 25%

P(positive | sick) = 50 / (50 + 10) = 50/60 ≈ 83.3%

P(sick | positive) = 25% ≠ P(positive | sick) = 83.3%. Base rate matters!

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Example 3 — Cards: Ace Given Red Card

P(Ace ∩ Red) = 2/52, P(Red) = 26/52

P(Ace | Red) = (2/52) ÷ (26/52) = 2/26 = 1/13 ≈ 7.69%

Note: same as unconditional P(Ace) = 4/52 = 1/13. Ace and Red card color are independent!

P(Ace | Red) = 7.69% = P(Ace). Knowing the card is red gives no info about whether it’s an ace.

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Example 4 — Bayes’ Theorem: Spam Filter

P(spam) = 20%, P(contains “free” | spam) = 60%, P(contains “free” | not spam) = 5%

P(“free”) = 0.60×0.20 + 0.05×0.80 = 0.12 + 0.04 = 0.16

P(spam | “free”) = (0.60×0.20) / 0.16 = 0.12 / 0.16 = 75%

P(spam | contains “free”) = 75%. Seeing the word “free” triples the spam probability from 20% to 75%.

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Frequently Asked Questions

What is conditional probability?+

Conditional probability P(A|B) is the probability that event A occurs given that event B has already occurred. It is calculated as P(A|B) = P(A∩B) / P(B), where P(A∩B) is the probability both events occur together (joint probability) and P(B) is the probability of the condition (marginal probability). It measures how likely A is in the restricted sample space where B is true.

What is the formula for conditional probability?+

P(A|B) = P(A∩B) / P(B). This requires P(B) > 0 (you cannot condition on an impossible event). Rearranging gives the multiplication rule: P(A∩B) = P(A|B) × P(B). Example: P(rain ∩ cloudy) = 0.30, P(cloudy) = 0.40 → P(rain | cloudy) = 0.30/0.40 = 75%.

What is the difference between P(A|B) and P(B|A)?+

What is Bayes' theorem?+

Bayes' theorem states: P(A|B) = P(B|A) × P(A) / P(B). It lets you update the probability of A after observing B. P(A) is the prior probability; P(A|B) is the posterior. Example: P(disease) = 0.01 (prior), P(positive test | disease) = 0.95, P(positive test | no disease) = 0.05. P(disease | positive test) = (0.95×0.01) / (0.95×0.01 + 0.05×0.99) ≈ 16.1%.

How do you calculate conditional probability from a contingency table?+

In a 2×2 table with cells: A∩B = a, ¬A∩B = b, A∩¬B = c, ¬A∩¬B = d: P(A|B) = a/(a+b) (of the B column, what fraction is A?); P(B|A) = a/(a+c) (of the A column, what fraction is B?). Row and column totals give marginal probabilities. The total of all cells = n (total observations).

What is the multiplication rule for probability?+

The multiplication rule states: P(A∩B) = P(A|B) × P(B) = P(B|A) × P(A). It gives the probability of both events occurring together. For independent events, P(A|B) = P(A), simplifying to P(A∩B) = P(A) × P(B). Example: drawing two aces without replacement: P(A₁∩A₂) = P(A₂|A₁) × P(A₁) = (3/51) × (4/52) ≈ 0.45%.

What does it mean for two events to be independent?+

Events A and B are independent if P(A|B) = P(A) - knowing B occurred gives no information about A. Equivalently: P(A∩B) = P(A) × P(B). If two fair dice are rolled, the result of die 1 is independent of die 2. But if a card is drawn and not replaced, the second draw depends on the first (dependent events).

What is base rate neglect?+

Base rate neglect occurs when people ignore the prior probability P(B) and focus only on P(A|B). Classic example: a test for a rare disease (prevalence 0.1%) has 99% sensitivity and 99% specificity. Most people say P(disease | positive) ≈ 99%, but the correct answer is about 9%: most positives are false positives because the disease is so rare. Always apply Bayes' theorem when evaluating test results.

How is conditional probability used in machine learning?+

Conditional probability is fundamental to machine learning: Naive Bayes classifiers use P(class | features) ∝ P(features | class) × P(class). Decision trees split on features that maximize information gain (related to conditional entropy). Logistic regression models P(Y=1 | X). Hidden Markov Models use transition probabilities P(state_t | state_{t-1}). Almost every probabilistic model in ML is built on conditional probability.

What is the law of total probability?+

If B₁, B₂, …, Bₙ are mutually exclusive and exhaustive events (they partition the sample space), then P(A) = Σ P(A|Bᵢ) × P(Bᵢ). This lets you compute an overall probability by conditioning on each partition. Example: P(disease) = P(disease|positive)×P(positive) + P(disease|negative)×P(negative). The law of total probability is the denominator in Bayes' theorem.

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📌 Quick Tips

💡P(A|B) ≠ P(B|A) in general. This is the most common mistake: 'the probability a patient has cancer given a positive test' is NOT the same as 'the probability of a positive test given cancer.' Bayes' theorem shows you how to go from one to the other.

💡If P(A|B) = P(A), then A and B are statistically independent - knowing B gives no information about A. Equivalently, P(A∩B) = P(A)×P(B) for independent events.

💡In the 2×2 contingency table mode: row totals give the marginal distribution of B; column totals give the marginal of A. The table cells are joint frequencies.

💡P(A∩B) cannot exceed P(B) or P(A). If it does, there is an error in your inputs.

💡Base rate neglect is a common cognitive bias: people ignore the low prior probability P(B) when computing P(A|B). Always consider how common B is in the population.

	B occurred	B did not occur
A occurred
A did not occur