1.5

1. 1.5 Conditional Probability Conditional Probability The multiplication rule

2. Definition 1.12 The conditional probability of an event A given than an event B has already occurred is given by Solution: Suppose that we roll a fair six-sided die and note the score obtained. Let A = the event that the outcome is > 3 and B= the event that the outcomeis an even number. What is the conditional probability that B occurs given that A has occurred? provided that P(B)>0

3. arts on time is P(D)=0.83; and arrives on time is given that it departed on time, given that it has arrived on time. Solution: Example 1.13 The probability that a regularly scheduled flight dep- the probability that it arrives on and the probability that it departs time is P(A)=0.82; Find the probability that a plane (a) arrives on time and (b) departed on time

4. occur, then Theorem 1.2 If in an experiment the events A and B can both Thus the probability that both A and B occur is equal to the probability that A occurs multiplied by the probability that B occurs, given that A occurs. Since the events are equivalent, it follows from Theorem 1.2 that we can also write In other words, it does not matter which event is referred to as A and which event is referred to as B.

5. proof: Following theorem generalizes these results to n events Theorem 1.3(The multiplication rule) If then

6. Solution: Example 1.14 Suppose that we have a fuse box containing 20 fuses, which 5 are defective, if 2 fuses are selected at random of and removed from the box in succession without replacing the first, what is the probability that both fuses aredefe- ctive?

7. Solution: Example 1.15 One bag contains 4 white balls and 3 black balls, and a second bag contains 3 white balls and 5 black balls. One ball is drawn from the first bag and placed unseen in thesecond bag. What is the probability that a ball now drawn from the second bag is black?

8. then Solution: We denote the conditional probability that A occurs give that B has occurred by

9. Solution: a. The probability that a plane arrives on time given that it departed on time is b. The probability that a plane departed on time given that it has arrived on time is

10. Proof:

11. then we interpretas the event that A occ- We shall let A be the event that the first Solution: fuse is defective and B the event that the second fuse is defective; urs, and then B occurs after A has occurred. The probability of first removing a defective fuse is removing a second defective 1/4; then the probability of Hence fuse from the remaining 4 is 4/19.

12. the drawing of a black ball from bag 1, a black ball from 61 Solution: Let B1, B2, and W1 represent, respectively, bag 2, and a white ball from bag 1. We are interested in the union of the mutually exclusive events The various possibilities and their probabilities are illustrated in Figure 1.2 .