Multiplication Rule

Independant Events

Two events A and B are said to be independant if occurance of one event does not change the probability of other.
For Example

• Owning a dog and winning a lottery.
• Liking football and being a doctor.

Multiplication rule for Independant Events

If events A and B are independant, probability of both occuring is given by
\(P(A \cap B) = P(A) * P(B)\)

Problem

What is the probability of throwing a dice twice and getting both 6s?
Let A be the event of throwing dice the first time and B be the event of throwing the dice for the second time. Now outcome of event B is not dependant on the outcome of event A, hence A and B are independant.
Now, \(P(A) = P(B) = \frac{1}{6}\)

\(P(A \cap B) = P(A) * P(B) = \frac{1}{6}*\frac{1}{6} = \frac{1}{36}\)

Multiplication rule for Dependant Events

If events A and B are dependant on each other, probability of both occuring is given by
\(P(A \cap B) = P(A) * P(B | A)\), where P(B|A) is the probability of B occuring given that event A has already occured. We will learn more about this in the next section.