1. C3: Conditional Probability And Independence
CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics 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
Provided P(C) > 0
3.2 – Multiplication Rule
For any events A and C:

3. 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)
What is P(Rc| L)?

4. Show that P(A | C) + P(Ac | C) = 1.
Hence the rule
P(Ac) = 1 – P(A)
also holds for conditional probabilities.

5. 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:
Or equivalently expressed as:

6. 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

7. 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.

8. 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.

9. 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.