1. C3: Conditional Probability And Independence
MATH 3033 based on Dekking et al. A Modern Introduction to Probability and Statistics Slides by James Connolly Format by Tim Birbeck Instructor Longin Jan Latecki
C3: Conditional Probability And Independence

2. 3.1 – Conditional Probability
Conditional Probability: the probability that an event will occur, given that another event has occurred that changes the likelihood of the event
Example: If event L is “person was born in a long month”, and event R is “person was born in a month with the letter ‘R’ in it”, then P(R) is affected by whether or not L has occurred. The probability that R will happen, given that L has already happened is written as:
P(R|L)

3. 3.1 – Conditional Probability
Provided P(C) > 0
3.2 – Multiplication Rule
For any events A and C:

4. 3.3 – Total Probability & Bayes Rule
The Law of Total Probability
Suppose C1, C2, … ,CM are disjoint events such that
C1 U C2 U … U CM = Ω. The probability of an arbitrary
event A can be expressed as:

5. 3.3 – Total Probability & Bayes Rule
Suppose the events C1, C2, … CM are disjoint
and C1 U C2 U … U CM = Ω. The conditional probability of Ci, given an arbitrary event A, can be expressed as: or

6. 3.4 – Independence Definition:
An event A is called independent of B if:
That is to say that A is independent of B if the probability of A occurring is not
changed by whether or not B occurs.

7. 3.4 – Independence Tests for Independence
To show that A and B are independent we have to prove just one of the following:
A and/or B can both be replaced by their complement.

8. 3.4 – Independence Independence of Two or More Events
Events A1, A2, …, Am are called independent if:
This statement holds true if any event or events is/are replaced by their
complement throughout the equation.