Practical Work 6 — Combinatorics and Probability #

Sixth practical class of Computer Discrete Mathematics. Working by hand, you count the outcomes of a choice (arrangements, combinations, multisets) and use those counts to find probabilities — of independent draws, of conditional events, and of “at least one” via the complement.

At a glance #

Contents #

Conventions #

• Language — all work is written in English.
• By hand. Set up each count or probability from a formula, show the substitution, and give the number; leave very large results as a product and its value.
• Notation is fixed and self-contained. $n!$ is the factorial; $P(n,k)=n!/(n-k)!$ is the number of arrangements ($k$-permutations); $C(n,k)=\binom{n}{k}$ is the number of combinations; $P(A)$ (upright, with an event inside) is a probability. All are defined in 2method.md.
• Probabilities are given as an exact fraction and a decimal (e.g. $\tfrac{1}{9}\approx0.111$).
• Every task has a full solution. 3classroom.md gives each task together with a complete worked answer.

Summary #

Combinatorics answers how many? and probability answers how likely? — and the second is built on the first. Every count reduces to two ideas: multiply choices made in sequence (the product rule) and decide whether order matters and whether repetition is allowed, which selects one of four formulas — arrangements $P(n,k)$, tuples $n^k$, combinations $C(n,k)$, or multisets $C(n{+}k{-}1,k)$. For equally-likely outcomes, probability is just counting: favourable over total. Layer on three rules — the product rule for independent trials, conditional probability when one event informs another, and the complement ($P(A)=1-P(\overline A)$) for “at least one” — and the nine tasks of this class are all within reach.

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