Combinatorics and Probability: Counting Rules and Chance
What you must understand
You need the multiplication and addition rules, permutations, combinations, and the basic definition of probability as favourable outcomes over total equally likely outcomes. Know the difference between ordered and unordered selections, with and without repetition, and be able to compute probabilities of simple and compound events, including complementary events.
Traps that appear in questions
- Order matters or not? Arranging 3 of 5 people in a line is a permutation (order matters); choosing 3 of 5 for a committee is a combination (order does not).
- With or without replacement? Drawing two cards without replacement changes the denominator on the second draw.
- Combinations formula. (\binom{n}{k}=\dfrac{n!}{k!(n-k)!}), and (\binom{n}{k}=\binom{n}{n-k}).
- Complement. (P(\text{not }A)=1-P(A)); often the fastest route.
- Independent events. (P(A\text{ and }B)=P(A)\cdot P(B)) only when independent.
Worked example
A bag contains 4 red and 6 blue balls. Two are drawn without replacement. What is the probability both are red? First draw: (4/10). Second draw (one red removed): (3/9). Product: (\dfrac{4}{10}\cdot\dfrac{3}{9}=\dfrac{12}{90}=\dfrac{2}{15}).
Rule of thumb: before computing, ask two questions - "does order matter?" and "is there replacement?" The answers determine the formula.