In previous sections, learned to calculate probabilities using the multiplication rule for events with unequal likelihoods.
Current focus: deriving a rule for calculating OR probabilities.
Understanding the OR Operator
Definition: OR between two events (A or B) includes all outcomes in either A or B.
Venn Diagram Representation:
Events A and B depicted in a sample space.
A or B: all outcomes in A, all in B, and those common to both.
Calculating Probability of A or B
Objective: Find probability of the union of events A or B.
Step-by-Step Process
Sum the Probabilities:
Begin by considering all outcomes in A.
Add outcomes from B.
This approach almost covers A or B but has an issue.
Overcounting Issue:
Adding probabilities of A and B results in double counting the intersection (common outcomes of A and B).
Correction for Overcounting:
Intersection (common region) is counted twice:
Once when counting A.
Again when counting B.
Adjust by subtracting the probability of the intersection once.
Addition Rule
Formula:
Probability of A or B = P(A) + P(B) - P(A and B)
Explanation: Ensures that common outcomes are only counted once.
Conclusion
Understanding this rule helps accurately calculate probabilities involving the union of two events by considering both individual probabilities and their overlap.