Conditional Probability

1. Conditional Probability Defining Conditional Probability The Multiplication Rule Independence of Events

2. Conditional Probability We begin with a simple example: • Toss a balanced die once and record the number on the top face. • Let E be the event that a 1 shows on the top face. • Let F be the event that the number on the top face is odd. • What is P(E)? • What is P(E) if we are told that the number on the top face is odd, that is, we know that the event F has occurred?

3. Conditional Probability • Key idea: The original sample space no longer applies. • The new or reduced sample space is S={1, 3, 5} • Notice that the new sample space consists only of the outcomes in F. • P(E occurs given that F occurs) = 1/3 • Notation: P(E|F) = 1/3

4. Conditional Probability Definition. If A and B are events and P(B)≠0, then we define the conditional probability of A given B by

5. Conditional Probability A B S

6. Example One • Suppose that a balanced die is tossed once. The relevant sample space is S={1, 2, 3, 4, 5, 6} • Let E be the event that we observe a 1. Since each element in S has an equal probability of occurring we have the unconditional probability

7. Example One • Define O to be the event that we observe an odd number. Then the unconditional probability is given by • Now, what is the probability of E given that O has occurred? Or, we might ask, what is the probability of Ogiven that E has occurred?

8. Example One • Symbolically, we write or to represent the two previous questions. In both cases, our original sample space S is not longer relevant. Let’s examine each case separately.

9. Example One • Consider • The new sample space is S ’={1,3,5} • So,

10. Example One • Using the definition we find • Similarly, we have

11. Conditional Probability Conditional probability problems lend themselves to several methods of solution. We can • Apply the definition • Construct Venn Diagrams • Construct Frequency (Contingency) Tables • Construct Tree Diagrams

12. Example Two • The probability that event A occurs is .63. The probability that event B occurs is .45. The probability that both events occur is .10. • Find the following probabilities by using the definition and Venn Diagrams: a. P(A|B); b. P(B|A); c. P(AC|B)d. P(BC|A); e. P(AC|BC)

13. Example Three – Using Tables Consider the experiment of rolling a fair die twice. All outcomes in S are equally likely.

14. Let E be the event that the sum of the faces is even Let Tbe the event that the second die is a 2 Find 1. P(T| E) 2. P(E |T) Example Three

15. Example Three • One way of doing this is to construct a table of frequencies:

16. Example Three • Let’s try it to find P(T| E) and P(E | T) Remember:

17. Multiplication Rule Using our definition of conditional probability we can perform the simple algebra of multiplying both sides of the equation by P(B) to get This is known as the multiplication rule of probabilities and provides another way at getting to the joint probability of A and B.

18. Example Four – Using Trees If a fair coin is flipped three times, what is the probability that it comes up tails at least once given: 1. No information at all. 2. All three coins produce the same side.3. It comes up tails at most once.4. The third flip is heads.

19. Example Five – Trees • Three manufacturing plants A, B, and C supply 20, 30 and 50%, respectively of all shock absorbers used by a certain automobile manufacturer. Records show that the percentage of defective items produced by A, B and C is 3, 2 and 1%, respectively. What is the probability that a randomly chosen shock absorber installed by the manufacturer will be defective? • What is the probability that the part came from manufacturer A, given that the part was defective? • What is the probability that the part came from B, given that the part was not defective?

20. Independent Events If the probability of the occurrence of event A is the same regardless of whether or not an outcome B occurs, then the outcomes A and B are said to be independent of one another. Symbolically, if then A and B are independent events.

21. Independent Events Recalling that then we can also state the following relationship for independent events: if and only ifA and B are independent events.

22. Example Six • A fair coin is tossed twice. Find the probability that a. the second toss is a head; b. the second toss is a head given that the first toss is a head.

23. Example Seven A card is drawn from a standard deck of 52 cards and then a second card is drawn without replacing the first card. Let A be the event that a second card is a spade and let B be the event that the first card is a spade. Determine whether A and B are independent events.

24. Example Eight You throw two fair die, one green and one red and observe the numbers. Decide which pairs of events, A and B, are independent:1. A : the sum is 5B : the red die shows a 22. A : the sum is 5B : the sum is less than 43. A : the sum is evenB : the red die is even

25. Independence The notion of independent events can be extended to conditional probabilities. If E, F, and G are three events, then E and F are independent, given that G has happened, if

26. Independence Likewise, events E, F, G are independent, given that an event H has happened, if

27. Example Nine • In the manufacture of light bulbs, filaments, glass casings and bases are manufactured separately and then assembled into the final product. Past records show: 2% of filaments are defective, 3% of casings are defective and 1% of bases are defective. • What is the probability that one bulb chosen at random will have no defects?