4.4: Conditional Probability and Independence

Question 1

What is conditional probability, and how is it denoted in probability theory?

Answer 1

Conditional probability is the probability of an event A occurring given that another event B has already occurred.

It is denoted as P(A∣B) and is read as “the probability of A given B.”

Question 2

Why is it necessary to obtain a more general formula for conditional probability?

Answer 2

Obtaining a more general formula for conditional probability is necessary because the reduced sample space approach demonstrated earlier to find conditional probabilities might not give accurate results when the sample space outcomes are not equally likely.

By expressing conditional probabilities in terms of

P(A), P(B), and P(A∩B),

a more general and applicable formula is derived for conditional probability that is valid for any sample space.

This ensures accurate calculations regardless of the distribution of outcomes.

Question 3

What is the general formula for conditional probability P(A∣B), and how is it defined?

Answer 3

Question 4

What is the general formula for conditional probability P(B∣A), and how is it defined?

Answer 4

Question 5

What are the equations derived from the multiplication rule for probabilities?

Answer 5

Question 6

What is the General Multiplication Rule?

Answer 6

Question 7

What is the definition of independent events, and how is independence between two events A and B determined?

Answer 7

Two events A and B are considered independent if and only if

P(A∣B)=P(A)

or, equivalently,

P(B∣A)=P(B).

In simpler terms, events A and B are independent if the occurrence (or non-occurrence) of one event does not influence the probability of the other event.

This means that knowing the outcome of one event does not provide any information about the likelihood of the other event happening

Question 8

What is the multiplication rule for independent events, and how is it expressed mathematically?

Answer 8