# Counting Sequences

**Introduction**

To calculate the probability of any event, we first need to **count **the number of all possible outcomes, and the number of favorable outcomes.

Counting efficiently becomes essential to process large amounts of statistical data. The number of outcomes of any event is written as .

Example: Let E be an event “month of a year”.

Since there are 12 months in a year, .

**The Basic Principle of Counting**

The fundamental/ basic rule of counting states that if we have outcomes associated to event , and outcomes of event , then there will be total outcomes in event and combined.

Note that the order in which these experiments are performed does not affect the total number of outcomes.

The basic rule of counting is also known as the **Multiplication Rule**.

To count the number of outcomes in such cases, we often use a *tree diagram*. We also use tree structures to often get the complete sample space for the combination of two events.

Example: Let’s say we have two events and .

: Tossing an unbiased coin.

: Rolling a fair dice.

We can sense here, and .

By using the fundamental principle for counting, we can say that the total number of outcomes for events and combined will be .

**Replacements and Ordering**

**Replacement** conditions play a crucial role in any counting problem.

There are two such conditions possible for any given counting problem.

1. Sampling **with replacement**.

2. Sampling **without replacement**.