An event and its complement partition the sample space
E and not E have no common outcome and together cover S. Therefore P(E)+P(not E)=1, often making indirect counting shorter.
Learning goals
- Use the complement rule and solve complement ratios.
- Choose direct counting or complement efficiently.
- Solve and validate multi-part real-life probability problems.
Concept lesson
Probability total and bounds
All elementary outcome probabilities sum to one; an event probability and its complement also sum to one, with each value from zero to one.
Complement rule
P(not E)=1-P(E). Subtract the event probability from one, not take its reciprocal, and verify the two values sum exactly to one.
Direct, complement, union and intersection
Use complement when the excluded set is shorter. For A or B, form the union without double-counting their intersection; for A and B, keep only outcomes satisfying both.
At least one head by complement
For two coins, complement of at least one head is no head={TT}. Hence P(at least one head)=1-1/4=3/4; direct listing gives the same result.
Complement ratio
If P(E):P(not E)=a:b, write probabilities as ak and bk. Since their sum is one, k=1/(a+b), giving a/(a+b) and b/(a+b).
Or versus and in numbered cards
List outcomes satisfying each number property. For or combine unique outcomes; for and retain only the intersection. Then divide by the unchanged total card count.
Complement-pair check
For weather, winning, defective items and success events, write the exact negated statement, subtract from one and add the pair back to verify.
Set-language filter
Translate at least, at most, not, or, and and neither into explicit subsets; remove duplicates from unions and compare direct with complement counts.
Multi-part probability ledger
Keep one sample denominator, list each favourable subset separately, note complement or overlap relationships, and check exhaustive disjoint parts sum to one.
Inverse category and complement chain
Use known category probabilities to find a remaining probability, use a supplied count to recover total population and all category counts, then validate that counts and probabilities exhaust the sample space.
Complement/application answer protocol
State the event relation, preserve one sample denominator, show direct or complement counts, answer each part and verify complement/exhaustive totals.
Complement and application mastery
Solve complement, complement-ratio, overlapping-condition and multi-part population problems with exhaustive probability checks.