October 16, 2000

Conditional probability

Definition For any two events A and B with P(B)>0, the conditional probability of A given that B has occurred is defined by P(A|B)=P(AÇB)/P(B)

Multiplication rule. P(AÇB)=P(A|B)P(B)

Exercise 1. A hiker leaves the point O shown in Figure 1, choosing one of the roads OB1, OB2, OB3 or OB4 at random. At each subsequent crossroads he again chooses a road at random. What is the probability that the hiker arriving at point A?

O

B3

B2 B4

B1

A

Tree Diagrams

The Law of Total probability, Bayes' Theorem

Definition. The events A1, A2,…,An are mutually exclusive if no two have any two common outcomes.

Definition. The events A1, A2,…,An are exhaustive if one Ai must occur, so that A1È…ÈAn=S.

The Law of Total Probability

If A1, A2,…,An are mutually exclusive and exhaustive events, then, for any other event B,

P(B)=P(B|A1)P(A1)+…+P(B|An)P(An)

Bayes' Theorem

If A1, A2,…,An are mutually exclusive and exhaustive events with P(Ai)>0 for I=1,…,n,, then, for any other event B, for which P(B)>0,

k=1,…,n

Exercise 2. One box contains six red balls and four green balls, and a second box contains seven red balls and three green balls. A ball is randomly selected from the first box and placed in the second box. Then a ball is randomly selected from the second box and places in the first box.

A. What is the probability that a red ball is selected from the first box AND a red ball is selected from the second box?

B. At the conclusion of the selection process, what is the probability that the numbers of red and green balls in the firs box are identical to the numbers at the beginning?

C. What is the probability that the ball selected from that second box is red?