Theorems

Theorem 2.8

The probability of $n_{0}$ failures and $n_{1}$ successes in $n = n_{0} + n_{1}$ independent trials is

P [E_{n_{0}, n_{1}}] = (\binom{n}{n_{1}}) (1 - p)^{n - n_{1}} p^{n_{1}} = (\binom{n}{n_{0}}) (1 - p)^{n_{0}} p^{n - n_{0}}

Abstract

The intuition here is that we

Theorem 2.9

P [E_{n_{0}, \dots, n_{m - 1}}] = (\binom{n}{n_{0}, \dots, n_{m - 1}}) p_{0}^{n_{0}} \dots p_{m - 1}^{n_{m - 1}} .

Notation

For a sequential experiment $E_{n_{0}, n_{1}}$ . The first $n_{0}$ denotes the number of failures, and $n_{1}$ denotes the number of successes such that there are $n = n_{0} + n_{1}$ trials.

Definitions

Tree Diagrams

Tree diagrams display the outcome of sub-experiments in a sequential experiment. The probability of an outcome in some sequential experiment is the product of the branches (each of which contain probabilities to the subsequent result) going from the root to the leaf.

Independent Trials

Independent trials are identical sub-experiments in a sequential experiment. The probability models of all sub-experiments are identical and the experiments are independent of the results of previous experiments. Sampling with replacement would fall under this category.