How to find probability using a Venn diagram (P(A or B) explained)

How to · Updated 2026-11-10

Count the outcomes in each region of the Venn diagram, then divide by the total: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Worked example, a survey shortcut and checks.

To find a probability from a Venn diagram, count how many equally likely outcomes fall in the region you want and divide by the total number in the universal set; for “A or B” use P(A ∪ B) = P(A) + P(B) − P(A ∩ B), so the overlap is not counted twice. The venn diagram calculator calculator shades any region you type, sorts real elements into the right parts of the picture and gives the counts, which you then turn into probabilities.

What you need

• The universal set U: every possible outcome, all equally likely. Picking a whole number from 1 to 30 gives 30.
• The events A and B, written as lists of outcomes (or as counts from a survey).
• The words of the question turned into set symbols:
  • “A or B” (or both) is A ∪ B, the union,
  • “A and B” is A ∩ B, the intersection,
  • “not A” is A′, the complement,
  • “A but not B” is A − B,
  • “neither” is (A ∪ B)′, which by De Morgan’s law equals A′ ∩ B′.

Step by step

The rule behind every answer is:

P(event) = number of outcomes in the event ÷ number of outcomes in U

and for two events,

P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

The minus is there because adding P(A) and P(B) counts the overlap twice.

With the tool:

1. Open the Venn diagram calculator and press 2 sets.
2. Type the event you want in the Expression box. The symbol keys add ∪, ∩, ′ and brackets, or you can type u, n and an apostrophe instead. The diagram shades that region, and the line under it names each shaded part.
3. Tick Put real elements in the sets. Type U, A and B as lists. Commas separate elements, and a dash gives a run, so 1-30 means every whole number from 1 to 30.
4. The result line writes your event out in full and gives n, its number of elements.
5. The counting facts below give n(U), n(A), n(B), n(A ∩ B), n(A ∪ B) with the subtraction shown, and the number in neither set.
6. Divide each count by n(U). The tool counts; turning a count into a probability is the one step you do yourself.

A worked example

A whole number from 1 to 30 is picked at random. A is “a multiple of 3” and B is “a multiple of 5”. Find the probability that the number is a multiple of 3 or 5, and the probability that it is neither.

• In the tool, type U as 1-30, A as 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 and B as 5, 10, 15, 20, 25, 30.
• The counts: n(A) = 10, n(B) = 6, and the overlap A ∩ B = {15, 30}, so n(A ∩ B) = 2.
• n(A ∪ B) = 10 + 6 − 2 = 14, so P(A ∪ B) = 14/30 = 7/15 ≈ 0.467.
• With probabilities instead: 10/30 + 6/30 − 2/30 = 14/30, the same answer.
• Neither: n(A′ ∩ B′) = 30 − 14 = 16, so P(neither) = 16/30 = 8/15 ≈ 0.533.
• Only a multiple of 3: type A − B as the expression. The result has 8 elements, so P = 8/30 = 4/15.

A survey shortcut. Many questions give counts, not lists: “In a class of 40, 22 play cricket, 15 play football and 7 play both.” Give each set a run of numbers that overlaps by the right amount. Type U as 1-40, A as 1-22 and B as 16-30. Then A has 22 members, B has 15, and they share 16 to 22, which is 7. The tool gives n(A ∪ B) = 30 and 10 in neither, so P(plays at least one) = 30/40 = 0.75 and P(plays neither) = 10/40 = 0.25.

Check your answer

• Regions add up. The four parts of a two-set diagram (A only, B only, both, neither) must add to n(U). Here 8 + 4 + 2 + 16 = 30.
• Complements add to 1. P(A ∪ B) + P(neither) = 7/15 + 8/15 = 1.
• Size limits. P(A ∪ B) can never be smaller than the bigger of P(A) and P(B), nor larger than their sum. 14/30 lies between 10/30 and 16/30.
• Count a few by hand. Run down 1 to 30 and tick each multiple of 3 or 5. You should tick 14 numbers.

Questions people ask

When can I just add P(A) and P(B)? Only when A and B cannot happen together, so the overlap is empty. Rolling a 2 and rolling a 5 on one die is such a pair. Then P(A ∩ B) = 0 and the rule becomes a plain sum.

What about dice, coins and cards? Writing out U gets long there. The sample space builder builder lists every outcome of two dice, coins or a deck and shades the event for you; the counting rule is the same.

How do I handle “B given A” questions? Conditional probability looks only inside A: P(B | A) = n(A ∩ B) ÷ n(A). In the first example that is 2/10 = 0.2. For problems built in stages, a probability tree lays out each branch and its probability.