Section 7.1

Binomial and Geometric

Bernoulli counts and waiting times: binomial probabilities and conditionings, comparisons of independent geometric variables, memorylessness of the geometric.

Toolkit

A Binomial $(n,p)$ count $X$ records successes in $n$ independent Bernoulli $(p)$ trials; a Geometric $(p)$ variable $T$ counts trials to first success. Key facts:

\mathbb{P}(X=k)=\binom{n}{k}p^k(1-p)^{n-k},\qquad \mathbb{P}(T=k)=(1-p)^{k-1}p,\ k\ge1.

Independence/exchangeability of Bernoulli trials: conditioning on a block of outcomes leaves the remaining trials i.i.d. Bernoulli $(p)$ , so counts over disjoint trial sets are independent binomials.
Memorylessness: $\mathbb{P}(T>m+n\mid T>m)=\mathbb{P}(T>n)$ ; equivalently, given failures so far, the remaining wait is a fresh Geometric $(p)$ .
Tail: $\mathbb{P}(T>n)=(1-p)^n$ ; for independent geometrics, $\mathbb{P}(\max_i T_i\le n)=\prod_i(1-(1-p_i)^n)$ .
Conditioning on irrelevant outcomes (e.g. restricting to two of several equally likely categories) turns a mixed waiting-time problem into a simple geometric comparison via relative probabilities.
Sums of independent geometrics with the same $p$ are negative binomial; phase-splitting decomposes compound waits into independent geometric pieces.

Problems