Algebra of events
Events combine like sets: A ∪ B means A or B (or both), A ∩ B means both happen, and Ā means A does not happen.
Combining events
| Event | Symbol | Happens when |
|---|---|---|
| A or B | A ∪ B | A happens, or B happens, or both |
| A and B | A ∩ B | A and B both happen |
| Not A | Ā (also written A′) | A does not happen |
| A but not B (only A) | A ∩ B̄ | A happens and B does not |
| Neither A nor B | Ā ∩ B̄ | Neither event happens |
Results to remember
- A ∩ Ā = ∅ and A ∪ Ā = S.
- If A and B are mutually exclusive, A ∩ B = ∅.
- (A ∩ B) ∪ (A ∩ B̄) = A, and these two parts are mutually exclusive.
- So P(A ∩ B̄) = P(A) − P(A ∩ B) and P(Ā ∩ B) = P(B) − P(A ∩ B).
- P(Ā) = 1 − P(A).
How to draw a Venn diagram
- Draw a rectangle and write S in one corner. It stands for the sample space.
- Draw two overlapping circles inside it and label them A and B.
- The overlap is A ∩ B; both circles together make A ∪ B.
- The part of A outside B is A ∩ B̄; the part outside both circles is Ā ∩ B̄.
- Write each outcome in its correct region, filling the overlap first, then count.
Example 1: one die
A die is rolled. A: an even number; B: a multiple of 3. Find A ∪ B, A ∩ B, Ā and A ∩ B̄, and their probabilities.
- S = {1, 2, 3, 4, 5, 6}, A = {2, 4, 6}, B = {3, 6}.
- A ∪ B = {2, 3, 4, 6}, so P(A ∪ B) = 4/6 = 2/3
- A ∩ B = {6}, so P(A ∩ B) = 1/6
- Ā = {1, 3, 5}, so P(Ā) = 3/6 = 1/2
- A ∩ B̄ = {2, 4}, so P(A ∩ B̄) = 2/6 = 1/3
- Check: P(A) − P(A ∩ B) = 3/6 − 1/6 = 2/6 = 1/3
P(A ∪ B) = 2/3, P(A ∩ B) = 1/6, P(Ā) = 1/2, P(A ∩ B̄) = 1/3
Example 2: from given probabilities
P(A) = 0.6, P(B) = 0.5 and P(A ∩ B) = 0.3. Find P(only A), P(only B), P(Ā) and P(B̄).
- P(A ∩ B̄) = P(A) − P(A ∩ B) = 0.6 − 0.3 = 0.3
- P(Ā ∩ B) = P(B) − P(A ∩ B) = 0.5 − 0.3 = 0.2
- P(Ā) = 1 − 0.6 = 0.4
- P(B̄) = 1 − 0.5 = 0.5
P(only A) = 0.3, P(only B) = 0.2, P(Ā) = 0.4, P(B̄) = 0.5
In a class survey, 'likes idli or dosa' is a union and 'likes both' is an intersection. A Venn diagram sorts such survey answers quickly.
1 marks: “If A and B are mutually exclusive events, what is A ∩ B?” The empty set, ∅.
2 marks: “If P(A) = 0.45 and P(A ∩ B) = 0.15, find P(A ∩ B̄) and P(Ā).” P(A ∩ B̄) = 0.45 − 0.15 = 0.30. P(Ā) = 1 − 0.45 = 0.55.
5 marks: “A number is chosen at random from 1 to 20. A: the number is even; B: the number is a multiple of 3. Find P(A ∩ B), P(A ∪ B), P(Ā) and P(A ∩ B̄).” n(S) = 20, n(A) = 10, n(B) = 6, A ∩ B = {6, 12, 18}. P(A ∩ B) = 3/20, P(A ∪ B) = 13/20, P(Ā) = 1/2, P(A ∩ B̄) = 7/20.
∪ looks like a cup that holds everything in A or B; ∩ is a cap over only the shared part.