2023-10-04
Use set process to calculate probability of event of interest
Calculate the probability of an event occurring, given that another event occurred.
Define keys facts for conditional probabilities using notation.
Clearly define your events of interest
Translate question to probability using defined events OR Venn Diagram
Ask yourself:
Are we sampling with or without replacement?
Does order matter?
Use axioms, properties, partitions, facts, etc. to define the end probability calculation into smaller parts
If probabilities are given to you, Venn Diagrams may help you parse out the events and probability calculations
If you need to find probabilities with counting, pictures or diagrams might help here
Write out a concluding statement that gives the probability context
(For own check) Make sure the calculated probability follows the axioms. Is is between 0 and 1?
Example 1
Suppose we randomly draw 2 cards from a standard deck of cards. What is the probability that we draw a spade then a heart?
Let
Let \(A =\) event \(1^{st}\) card is spades
Let \(B =\) event \(2^{nd}\) card is heart
Fact 1: General Multiplication Rule
\[\mathbb{P}(A\cap B)=\mathbb{P}(A)\cdot\mathbb{P}(B|A)\]
Fact 2: Conditional Probability Definition
\[\mathbb{P}(A|B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}\]
Fact 3
If \(A\) and \(B\) are independent events (\(A \unicode{x2AEB}B\)), then \[\mathbb{P}(A|B) = \mathbb{P}(A)\]
Fact 4
\(\mathbb{P}(A|B)\) is a probability, meaning that it satisfies the probability axioms. In particular, \[\mathbb{P}(A|B) + \mathbb{P}(A^C|B) = 1\]
Example 2
Two dice (red and blue) are rolled. If the dice do not show the same face, what is the probability that one of the dice is a 1?
Chapter 4 Slides