Visualize P(A), P(B), unions, intersections, and conditional probabilities. Built for statistics class and probability homework.
Enter event probabilities and choose an operation - union, intersection, conditional, complement, independence, and inclusion-exclusion. Built for statistics class, GMAT/SAT prep, and probability homework.
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Step-by-Step Solution
Probability can often feel like a collection of abstract formulas, but at its heart, it's about the relationship between events. A Probability Venn Diagram is the most effective way to turn those formulas into something you can actually see. By representing events as overlapping circles, you can instantly understand why P(A ∪ B) isn't just P(A) + P(B), but requires subtracting the intersection to avoid double-counting.
Whether you're a student tackling introductory statistics, a researcher mapping out joint distributions, or someone preparing for a standardized test like the GMAT or GRE, these diagrams provide the "Aha!" moment that algebra alone often misses.
This free online tool allows you to build custom 2-event and 3-event probability diagrams. You can input probabilities for every specific region-from the central triple intersection to the exclusive "only A" areas-and download your creation as a clean, high-resolution SVG or PNG for your notes or presentations.
Visualizing P(A∪B), P(A∩B), and P(A|B) is the fastest way to teach inclusion-exclusion. Probability Venns appear in nearly every introductory statistics textbook chapter on events.
Two- and three-event Venn problems are a staple of standardized test maths sections. Drawing one quickly is a learnable skill - practice with this tool and the formulas become reflexes.
“What’s the probability of drawing a red card or a face card?” A Venn shows that P(red ∪ face) = 26/52 + 12/52 − 6/52 = 32/52 in one glance.
Sensitivity, specificity, and Bayes-style problems often involve overlapping events: tested positive, has the disease, false positives. A probability Venn unpacks these.
Probability that a customer uses both Product A and B, given they’re a paying user. Conditional probability problems map perfectly onto Venn diagrams with overlap regions.
A 2-circle Venn is the cleanest way to introduce Bayes’ theorem. The formula P(A|B) = P(A∩B)/P(B) becomes geometric: the overlap divided by the size of B.
Every region of a probability Venn diagram corresponds to a formula. These are the four you’ll use 90% of the time.
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
The probability that A or B (or both) occurs. Add the two event probabilities and subtract the overlap once.
P(A ∩ B) = P(A) · P(B | A) = P(B) · P(A | B)
The joint probability that both A and B occur. If A and B are independent, this simplifies to P(A) · P(B).
P(A | B) = P(A ∩ B) / P(B)
The probability of A given that B has already happened. On a Venn diagram, this is the overlap area divided by the area of circle B.
P(A | B) = P(B | A) · P(A) / P(B)
Inverts the conditional probability. Critical for diagnostic testing, spam filters, and any problem where you need to reason backwards from observation to cause.
A and B are independent ⇔ P(A ∩ B) = P(A) · P(B)
If the joint probability equals the product of the individual probabilities, the events are independent. Otherwise, they’re dependent.
If A and B are mutually exclusive, P(A ∩ B) = 0
The circles don’t overlap. P(A ∪ B) simplifies to P(A) + P(B). Examples: rolling a 1 and rolling a 6 on the same die.
Problem: A card is drawn from a standard 52-card deck. What is the probability that it is either a heart or a face card (J, Q, K)?
Problem: 60% of customers buy product A, 40% buy product B, and 25% buy both. Given that a customer bought A, what is the probability they also bought B?
Interpretation: About 41.7% of customers who bought A also bought B.
Problem: A disease affects 1% of a population. A test is 95% accurate (sensitivity and specificity). If a patient tests positive, what is the probability they actually have the disease?
Surprise result:Even after testing positive, the patient has only a ~16% chance of actually having the disease — classic Bayesian counter-intuition.
Problem: A user opens an app daily. Probability they like feature A is 0.6, feature B is 0.5, feature C is 0.4. Pairwise overlaps are P(A∩B) = 0.3, P(A∩C) = 0.2, P(B∩C) = 0.15. P(A∩B∩C) = 0.1. What is the probability they like at least one feature?
Conditional probability P(A|B) is the ratio of the overlap area to the entire B circle. To show it on a Venn diagram, label the overlap with P(A∩B) and the rest of circle B with P(B only), then write the formula P(A|B) = P(A∩B)/P(B) below the diagram.
You can, but mutually exclusive events have no overlap by definition. The diagram will show two circles with the intersection labelled 0. For mutually exclusive events, P(A∪B) = P(A) + P(B) directly — no subtraction needed.
The probability that neither event occurs is P(A′ ∩ B′) = 1 − P(A∪B). You can include this as a note in the subtitle, or write it in the white space outside the circles.
Yes — if you include the “outside” region for events neither in A, B, nor C. The total area of the universal set always represents probability 1.
Yes — the diagram is rendered entirely in your browser. No probabilities you enter are uploaded anywhere.
Probability problems involving multiple events are where Venn diagrams really earn their keep. Once you can sketch the right diagram, the algebra usually falls out in a couple of lines. This tool gives you a clean, customizable canvas for building probability Venns — ideal for textbooks, lecture slides, GRE/GMAT prep, or any work where you need to communicate joint and conditional probabilities clearly.