Product Rule Principle in Probability

What is the fundamental concept behind the Sum Rule Principle?

If two events A and B are disjoint, what is the value of n(A ∩ B)?

Suppose an event can occur in 5 ways, and independently of this, a second event can occur in 3 ways. How many ways can the events occur together?

What is the condition for the Sum Rule Principle to be applied?

If there are 5 faculty members and 3 students, how many ways can a representative be chosen for a university committee if either a faculty member or a student can be chosen?

What is the formula for the number of ways two events A and B can occur together, if they are independent?

If two events A and B are not disjoint, what is the value of n(A ∪ B)?

What is the main difference between the Sum Rule Principle and the Product Rule Principle?

If there are 3 ways to choose a menu for a dinner party, and independently of this, there are 4 ways to choose a dessert, how many ways can the dinner party be planned?

What is the purpose of combinatorial analysis?

Product Rule Principle

If two events occur independently, then the total number of ways they can occur is the product of the number of ways each event can occur.
Formula: 𝑛( 𝐴× 𝐵 ) = 𝑛( 𝐴 ) × 𝑛( 𝐵 )
Example: 3 flights from California to France and 2 flights from France to Sri Lanka, total number of flight plans is 3 × 2 = 6.

Sum Rule Principle

If two events cannot occur simultaneously, then the total number of ways they can occur is the sum of the number of ways each event can occur.
Formula: 𝑛( 𝐴+ 𝐵 ) = 𝑛( 𝐴 ) + 𝑛( 𝐵 )
Example: 27 faculty members and 83 CS majors, total number of choices for a representative is 27 + 83 = 110.

Permutations

An arrangement of items in a particular order, where order matters.
Formula: 𝑛! / (𝑛−𝑟)!
Example: 10 people in a competition, top three can be ordered in 10 × 9 × 8 = 720 ways.
Permutations can be calculated using the fundamental counting principle or factorial notation.

Permutation Examples

3 people can be arranged in 3 × 2 × 1 = 6 ways.
10 people can be arranged in 10! / (10-3)! = 720 ways.

Permutation Practice

A lock can be opened in 30 × 29 × 28 = 24360 ways.
A club of 24 members can elect officers in 24 × 23 × 22 × 21 = 255,024 ways.

Combinatorial Analysis

Includes the study of permutations and combinations.
Concerned with determining the number of logical possibilities of some event without identifying every case.

Learn about the product rule principle in probability theory, including examples and formulas to calculate the number of ways events can occur independently.