Complementary Events
โFor 'at least one', subtract 'none at all' from 1.โ
The formula
P(Aแถ) = 1 โ P(A)How to read it: the probability that A does not happen is 1 minus the probability that A does happen
- A
- โ some event
- Aแถ
- โ the event that A does not happen (its complement)
- 1 โ P(A)
- โ the whole probability 1 minus the probability that A happens
The hook
A probability that's a pain to count directly โ like 'at least one' โ often cracks open the moment you flip it: 1 minus 'none at all'.
In plain words
The probability that something does NOT happen is the whole probability 1 minus the probability that it does.
The intuition
Happening and not-happening together always make 100% (=1). The two slices fill the whole circle, so once you know one, the other is just 1 minus it.
How it's built
An event A and its complement Aแถ never overlap (can't both happen) and together they cover everything. So P(A)+P(Aแถ)=1, and rearranged, P(Aแถ)=1โP(A).
Example
Probability of 'at least one heads' in three coin tosses: the complement is 'all tails' at 1/8. So 1 โ 1/8 = 7/8.
Common mistake
The complement isn't 'less than A' โ it's 'everything except A'. The opposite of 'at least one' is not 'at most one' but 'none at all'.
Where it's used
'At least ~' and 'at least once' problems are almost always faster via the complement. Especially powerful when flipping success and failure across many tries.
Where it came from
The idea that happening and not-happening sum to 1 fell out naturally from early probability, which treated the whole as 1 and split it into shares.
Prerequisites
Quick check
If the probability of rain tomorrow is 0.3, what is the probability that it does not rain?
- 0.3
- 0.5
- 0.7โ
- 1
Practice
If the probability of an event is 0.25, what is the probability that it does not happen?
Answer: 0.75
- P(happens) = 0.25
- 1 โ 0.25 = 0.75
Takeaway: P(not happening) = 1 โ P(happening).
Rolling one die, what is the probability of NOT getting a 1?
Answer: 3
- P(getting a 1) = 1/6
- 1 โ 1/6 = 5/6
Takeaway: 'Not' is the complement โ subtract from 1.
Using the complement, find the probability of 'at least one heads' in two coin tosses.
Answer: undefined
- Complement 'all tails' = 1/4
- 1 โ 1/4 = 3/4
Takeaway: For 'at least one', it's faster to count the opposite ('none at all').