L2 C3 S2

S2: Independent Events

In the previous example, the events of being female and wearing glasses are NOT independent. If a student is female, then P(G|F)=10/40=25%. If a student is male then P(G|M)=25/60=41.67%

The gender of the student gives information about whether or not he/she wears glasses -- Males are more likely to wear glasses. The reverse is also true. If a student wears glasses than P(F|G)=10/35 =2/7 and P(M|G)=25/35=5/7. He/she is much more likely to be male.

MULTIPLICATION LAW FOR INDEPENDENT EVENTS

Two events A and B are independent if the probability of their JOINT occurence is the PRODUCT of their separate probabilities.

Definition: Two events A and B are independent if knowledge of one gives NO INFORMATION about the other. In formulas, P(A|B)=P(A)

The conditional probability of A given B is the same as the original probability of A without conditioning. This would happen if the proportion of glass wearers among females is the SAME as the proportion of glass wearers among males. Using the formula for conditional probability, we see that A and B are independent if:

The Multipliclation Law is extremely helpful in allowing us to compute the probabilities of independent events. We will show how to apply it in some real world examples.