What it does
An event is a set of outcomes. Roll two dice and “the sum is 9 or more” is the 10 outcomes (3, 6), (4, 5), (4, 6) … (6, 6). Once events are sets, you can combine them like sets:
- A ∪ B (A or B): outcomes in A, in B, or in both;
- A ∩ B (A and B): outcomes in both;
- A′ (not A): every outcome that is not in A;
- A − B (A but not B), and the outcomes in exactly one of A and B.
This page draws all 36 outcomes of two dice as a 6 × 6 grid (or the 12 outcomes of a coin and a die). Cells in A are shaded blue, cells in B orange, cells in both half and half, and the cells of the event you choose are outlined. Underneath, the addition rule is worked with the actual counts:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
The − P(A ∩ B) is there because the overlap gets counted twice when you add P(A) and P(B). When the overlap is empty, A and B are mutually exclusive and the rule becomes plain addition.
It fits the Class 11 maths chapter on probability (events, the algebra of events, axiomatic probability), and builds on the classical probability of Class 9 and 10. Count simple events first with the sample space builder tool and see the sets as circles with the venn diagram calculator; for “given that” questions in Class 12, go on to the probability tree.
How to use it
- Choose the experiment: Two dice or A coin and a die.
- Pick Event A and Event B from the lists. Some events take a number k (the sum is k, the die shows k or more); a k box appears for those.
- Tap an Outline button: A ∪ B, A ∩ B, A′, A − B, B − A or “exactly one”. The outlined cells are counted in the last line of the box.
- Read P(A), P(B), P(A ∩ B) and the outlined event as counts out of n(S) and as fractions in lowest terms, then the addition rule with your numbers.
- Check it by experiment rolls 100, 1000 or 10 000 times and compares the share of rolls in A ∪ B with the rule.
In the dice grid the rows are the first die and the columns the second; each cell shows the sum of the two dice.
Worked example
The page opens with A = “the sum is 9 or more” and B = “at least one die shows 6”. The box reads P(A) = 10/36 = 5/18, P(B) = 11/36 and P(A ∩ B) = 7/36: seven cells, such as (3, 6) and (6, 6), are in both. The rule line says P(A ∪ B) = 10/36 + 11/36 − 7/36 = 14/36 = 7/18, and counting the outlined cells gives 14 as well. Because 7 outcomes are shared, A and B are not mutually exclusive.
Now change A to “the sum is k” with k = 7 and B to “a double”. The overlap is empty (a double always has an even sum), so the page says they are mutually exclusive and P(A ∪ B) = 6/36 + 6/36 = 1/3.
FAQ
What is the difference between mutually exclusive and exhaustive? Mutually exclusive events cannot happen together (no shared cells). Exhaustive events cover every outcome between them (no empty cells). Heads and tails are both.
Why does the simulation not give exactly the rule’s number? Random rolls wobble. With more rolls the relative frequency usually gets closer to the probability, which is what the rule describes.
Is anything stored? No. Everything runs in your browser. fibub is not part of NCERT; the topics follow the syllabus, but the words, pictures and examples are our own.
Where the numbers come from
- Sample spaces, events, the algebra of events and the addition rule of probability are standard school probability. Wording, pictures and examples are fibub's own; no textbook text, figures or exercises are used.
- Checked against a Python enumeration of all 36 outcomes of two dice (itertools.product with fractions.Fraction) and a seeded simulation of 20 000 rolls.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.