Theoretical Probability and Counting

Counting Principles

Fundamental Counting Principle

The fundamental counting principle states that if there are m ways to do one thing, and n ways to do another, then there are m * n ways of doing both.

Permutations

A permutation is the arrangement of objects in a specific order. The number of permutations of n distinct objects taken r at a time is given by the formula: nPr = n! / (n - r)!

Combinations

A combination is a selection of objects without considering the order. The number of combinations of n distinct objects taken r at a time is given by the formula: nCr = n! / (r! * (n - r)!)

Theoretical Probability

Sample Space

The sample space is the set of all possible outcomes of an experiment. It is denoted by S.

Event

An event is a subset of the sample space, representing the outcomes of interest.

Probability of an Event

The probability of an event A, denoted by P(A), is the ratio of the number of favorable outcomes to the total number of possible outcomes.

Study Guide

Fundamental Counting Principle

Use the fundamental counting principle to determine the number of outcomes for a given scenario.

Example

If there are 3 different shirts and 4 different pants, how many different outfits can be created?

Solution: By the fundamental counting principle, the number of different outfits is 3 * 4 = 12.

Permutations

Use the permutation formula to calculate the number of arrangements of objects.

Example

How many different 3-letter arrangements can be made using the letters A, B, C, and D?

Solution: The number of permutations is 4P3 = 4! / (4 - 3)! = 4 * 3 * 2 = 24.

Combinations

Use the combination formula to calculate the number of selections of objects.

Example

A committee of 5 members is to be formed from a group of 10 people. How many different committees can be formed?

Solution: The number of combinations is 10C5 = 10! / (5! * (10 - 5)!) = 252.

Theoretical Probability

Calculate the probability of an event using the ratio of favorable outcomes to total outcomes.

Example

What is the probability of rolling a 3 on a fair six-sided die?

Solution: The probability is P(3) = 1/6.

By mastering the fundamental counting principle, permutations, combinations, and theoretical probability, you will have a solid foundation in understanding and calculating probabilities in various scenarios.