Lesson 3 of 11 · 7 min
Algebra of events
NCERT §14.1.3
The dice stall pays out on two separate signs: 'Prime wins a pen' and 'Odd wins a balloon'. A visitor rolls a 3 and claims both. Another rolls a 2. What does each sign's event look like, and how do the two combine?
The lesson in notes
In short
Events are sets, so they combine with the set operations studied in the Sets chapter: complement, union, intersection and difference.
Complement, the event 'not A': A′ = S − A = {ω : ω ∈ S, ω ∉ A}. It occurs exactly when A does not.
Three coins: A = {HTH, HHT, THH} is 'only one tail appears'. Its complement is A′ = {HHH, HTT, THT, TTH, TTT}; an outcome such as HTT means 'not A' happened.
'A or B' = A ∪ B: the outcome lies in A, in B, or in both. 'A and B' = A ∩ B: the outcome lies in both.
'A but not B' = A − B = A ∩ B′: the outcome lies in A and outside B.
Die thrown twice: A = 'first throw is six' has 6 outcomes, and B = 'sum at least 11' has just three, (5,6), (6,5) and (6,6). Only (6,5) and (6,6) lie in both, so A ∩ B is the event 'first throw six and sum at least 11'.
NCERT's Example 1: on a die let A = prime = {2, 3, 5} and B = odd = {1, 3, 5}. Then A ∪ B = {1, 2, 3, 5}, A ∩ B = {3, 5}, A − B = {2} and A′ = {1, 4, 6}.