2.4 Conditional Probability

Transcription

1 2.4 Conditional Probability The probabilities assigned to various events depend on what is known about the experimental situation when the assignment is made. Example: Suppose a pair of dice is tossed. Given that the sum of the face value is 6, find the probaility that the face value on each die is 3. 1

2 Let s Make a Deal! Imagine that the set of Monty Hall s game show Let s Make a Deal has three closed doors. Behind one of these doors is a car; behind the other two are goats. The contestant does not know where the car is, but Monty Hall does. The contestant picks a door and Monty opens one of the remaining doors, one he knows doesn t hide the car. If the contestant has already chosen the correct door, Monty is equally likely to open either of the two remaining doors. After Monty has shown a goat behind the door that he opens, the contestant is always given the option to switch doors. What is the probability of winning the car if she stays with her first choice? What if she decides to switch? 2

3 P (A B): the conditional probability of event A given that the event B has occured. Venn diagram: For any two events A and B with P (B) > 0, the conditional probability of A given that B has occured is defined by P (A B) = P (A B) P (B) 3

4 Example: The probability that a regularly scheduled flight departs on time is P (D) = 0.83; the probability that it arrive on time is P (A) = 0.82; and the probability that it departs and arrives on time is P (D A) = Find the probability that a plane a. arrives on time given that it departed on time, b. departed on time given that it has arrived on time. 4

5 The Multiplication Rule: P (A B) = P (A B)P (B) = P (B A)P (A) This rule is important because it is often the case that P (A B) is desired, whereas both P (B) and P (A B) can be specified from the problem description. Example: Select 2 cards at random(w/o replacement) in a pack of cards. A=selecting an ace on the first draw, B=selecting an ace on the second draw What is P (A B)? 5

6 Example: Exercise 2.51 (P75 of the 7th edition) One box contains 6 red balls and 4 green balls, and a second box contains 7 red balls and 3 green balls. A ball is randomly chosen from the first box and placed in the second box. Then a ball is randomly selcted from the second box and placed 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 first box are identical to the numbers at the beginning? 6

7 The computation of a posterior probability from given prior probablity P (A i ) and conditional probabilities P (B A i ) occupies a central position in elementary probability. Events A 1,..., A k are mutually exclusive if no two have common outcomes. The events are exhaustive if one A i must occur, so that A 1... A k = S 7

8 The Law of Total Probability Let A 1,..., A k be mutually exclusive and exhaustive events. Then for any other event B, P (B) = P (B A 1 )P (A 1 ) P (B A k )P (A k ) = k i=1 P (B A i )P (A i ) 8

9 Example: In a certain assembly plant, three machines, B 1, B 2, and B 3, make 30%, 45%, and 25%, respectively, of the products. It is known from past experience that 2%, 3%, and 2% of the products made by each machine, respectively, are defective. Now, suppose that a finished product is randomly selected. What is the probability that it is defective? 9

10 Example: You are given a chance to win 1M dollar. You are given two wood boxes, 10 black balls and 10 white balls, you can put the balls into the box however you want. You have to choose a box at random first, then you must choose a ball from that box at random. If the ball is black, you win 1M dollar; otherwise, you win nothing. How will you allocate the ball into the boxes? 10

11 Example: A communication source transmits symbols 0 s and 1 s independently with probability 0.6 and 0.4, respectively. Assume that the channel is noisy. At the receiver one obtain symbols of 0 s and 1 s but with the chance that any particular symbol was distorted at the channel is What is the probability of receiving a one? 11

12 Bayes Theorem Let A 1, A 2,..., A k be a collection of k mutually exclusive and exhaustive events with P (A i ) > 0 for i = 1,..., k. Then for any other event B with P (B) > 0, P (A j B) = P (A j B) P (B) P (B A j )P (A j ) = ki=1 P (B A i )P (A i ) for j = 1,..., k. 12

13 The proliferation of events and subscriptions in the above theorem can be a bit intimidating to probability newcomers. As long as there are relatively few events in the partition, a tree diagram can be used as a basis for calculating posterior probability without ever referring explicitly to Bayes theorem. 13

14 Example: In a certain region of the country it is known from past experience that the probability of selecting an adult over 40 years of age with cancer is If the probability of a doctor correctly diagnosing a person with cancer as having the disease is 0.78 and the probability of incorrectly diagnosing a person without cancer as having the disease is 0.06, a. what is the probability that a person is diagnosed as having cancer? b. What is the probability that a person diagnosed as having cancer actually has the disease? 14