1 Conditional Probabilities

Transcription

1 1 Conditional Probabilities We use a special notation of the probability to clarify the sample space( the set of all possibilities under consideration ) P (A S):= conditional probability of event A relative to the sample space S or the probability of A given S. In P (A S), usually, the sample space S is smaller than the original sample space. Example 1.1 Suppose that a consumer research organization has studied the service provided by the appliance repair persons in a certain city. Its findings are summarized in the following table: Good Service Poor Service Total Factory trained Not factory trained Total a. What is the probability of choosing one who provides good service if one of these repair persons is randomly selected? P (G) = 72 b. What is the probability of choosing one who is factory trained? P (F ) = 64 c. What is the probability of choosing one who provides good service and is factory trained? P (G F ) = 48 1

2 d. What is the probability of choosing one who provides good service among those who are factory trained? P (G F ) = 48 P (G F ) (= 64 P (F ) = ) Proposition 1.2 If P (B) is not equal to zero, then the conditional probability of A relative to B, namely, the probability of A given B, is P (A B) = P (A B) P (B) Example 1.3 A computer manufacturer feels that the probability is 0.75 that the computer chips( which are essential to fill an order for computers) will arrive on time, and the probability is 0.60 that the order for computers will be filled on time. If C is the event that the chips will arrive on time, and F is the event that the order for computers will be filled on time, then P (C) = 0.75 and P (F C) = What is the probability that the order will be filled on time given that the chips arrive on time? P (F C) = P (F C) P (C) = = 0.80 Example 1.4 The records of annual meetings of a certain corporation show that the probability is 0.30 that its stockholders will attend the annual meeting in person (instead of, say, submitting a proxy). The probability that a stockholder who attends the meeting will vote at the meeting is What is is the probability that a stockholder will attend the meeting in person and will vote at the meeting? So, 0.25 = P (V A) = P (A V ) P (A) = P (A V )

3 P (A V ) = Example 1.5 A refiner of gasoline feels that the probability is 0.80 that a tanker carrying crude oil needed to fill an order for gasoline will arrive on time, and the probability is 0.60 that the crude oil will arrive on time and the order for gasoline will be filled on time. What is the probability that the order for gasoline will be filled on time given that the tanker carrying the crude oil arrives on time? P (G C) = P (G C) P (C) = = 0.75 Independent Events Example 1.6 The probabilities that a student will get passing grades in algebra, in literature, or in both subjects are, respectively, P (A) = 0.75, P (L) = 0.84, and P (A L) = What is the probability that the student will get a passing grade in algebra given that he or she gets a passing grade in literature? P (A L) = P (A L) P (L) = = 0.75(= P (A)) The probability of event A is the same regardless of whether or not event L has occurred(occurs, or will occur). Definition: If P (A B) = P (A), then the event A is independent of event B. that is, 3

4 Event A is independent of event B if the probability of event A is not affected by the occurrence or nonoccurrence of event B. We say that two events are dependent events if they are not independent. Multiplication Rules: Proposition 1.7 (General multiplication rule) P (A B) = P (B) P (A B)(= P (A) P (B A) (The probability that two events will both occur is the product of the probability that one event will occur and the conditional probability that the other event will occur given that the first event has occurred(occurs, or will occur).) Example 1.8 A jury consists of nine person who are native born and three persons who are foreign born. If two of the jurors are randomly picked for an interview, what is the probability that they will both be foreign born? A; the event that the first juror is foreign born B; the event that the second juror is foreign born. Then, P (A) = 3 12, P (B A) = 2 (the probability that the second juror will 11 also be foreign born when the first juror picked is foreign born). P (A B) = P (A) P (B A) = 3 12 Proposition 1.9 (Special multiplication rule) If A and B are independent events, then P (A B) = P (A) P (B) 2 11 =

5 Example 1.10 If P (C) = 0.65, P (D) = 0.40, and P (C D) = 0.26, are the events C and D independent? P (C) P (D) = = 0.26 = P (C D) The two events are independent. 5