STAT 3401, Intro. Prob. Theory 16/22 Jaimie Kwon
1/19/2005
2.8 Two laws of probability
♦
Theorem 2.5. (The multiplicative Law of Probability) For two events A and B,
P(A∩B)=P(A)P(B|A) = P(B)P(A|B).
If A and B are independent, then
P(A∩B)=P(A)P(B)
à Proof: follows from the definitions
à Applying it multiple times, we can have results like
P(A1∩A2∩…∩Ak)=
P(A1)P(A2| A1)P(A1| A1∩ A2)…P(Ak|A1∩ A2∩… ∩ Ak-1)
♦
Theorem 2.6. (The additive Law of Probability) For two events A and B,
P(A∪B)=P(A)+P(B)-P(A∩B)
If A and B are mutually exclusive, (so P(A∩B)=0) then
P(A∪B)=P(A)+P(B)
à Proof: inspection of the Venn diagram
à P(A∪B∪C)=?
♦
Theorem 2.7. P(A) = 1-P(A’)
à Proof: use the previous theorem.
♦ HW: some of the exercises 2.66~85
♦ Keywords: the multiplicative & additive law of probability;
2.9 Calculating the probability of an event: the event-composition
method
♦ Example 2.17. Political composition of a city and their support of the bond issues
R vs. D (40% vs. 60%), P(F|R)=.7 and P(F|D)=.8. What is P(F)?
♦ Example 2.18. Birthday problem. We know P(A)=.5886. What’s P(B)=P(A’)?
♦
Steps for the event-composition method (use the multiplicative/additive laws)
à Define the experiment
à Visualize the nature of the sample points. Identify a few to clarify your thinking.
à Write an equation expressing the event of interest, say A, as a composition of two or more
events, using unions, intersections and/or complements. (*hardest*)
à Apply the laws of probability to the compositions to find P(A)
