Probability and counting
Probability and counting questions reward a clear sample-space decision. Decide whether order matters, whether selection is with or without replacement, and whether the question asks for a count or a probability.
Core rules
- Permutations count ordered arrangements: nPr=n!/(n-r)!. Combinations count unordered selections: nCr=n!/(r!(n-r)!).
- For events A and B, P(A∪B)=P(A)+P(B)-P(A∩B). If independent, P(A∩B)=P(A)P(B).
- Conditional probability: P(A|B)=P(A∩B)/P(B), provided P(B)>0. This changes the denominator to the conditioning event.
- Expected value: E(X)=Σ x P(X=x). It is a weighted average and need not be a possible value of X.
- Use complementary counting when direct cases are numerous, and split into mutually exclusive cases.
Worked example
A bag has 3 red and 2 blue balls. Two balls are drawn without replacement. Find the probability that they have the same colour. The total number of unordered pairs is 5C2=10. Same colour means two red or two blue: 3C2+2C2=3+1=4. Thus the probability is 4/10=2/5. Equivalently, P(RR)+P(BB)=(3/5)(2/4)+(2/5)(1/4)=6/20+2/20=8/20=2/5.
Traps
Do not use nCr when positions or order are distinct. Without replacement, denominators shrink after each draw. In conditional probability, using the original total instead of the condition's restricted sample space is a common error. Mutually exclusive events are not independent unless one has probability zero. Check whether cases overlap before adding probabilities.