Conditional Probability P(A) represents the probability assigned to A, it is the original or unconditional probability Sometimes there may be conditions “an event B has occurred” that affects the probability assigned to A The conditional probability of an event, A, given that an event B has happened is denoted P(AB) P(AB) is read “the probability that A occurs given that B has occurred”

Conditional Probability, con’t Given that B has occurred, the relevant sample space has changed, it is no longer S but consists only of the outcomes in B A has occurred if and only if one of the outcomes in their intersection have occurred For any events A and B with P(B)>0,

Conditional Probability, con’t

Conditional Probability, con’t We can solve these problems using several methods: Use the formula Venn Diagrams Frequency Tables Tree Diagrams

Example-using definition 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 by using the definition: P(AB) P(BA)

Example-Using Venn Diagram Suppose that A and B are events with probabilities: P(A)=1/3, P(B)=1/4, P(AB)=1/10 Find each of the following using a Venn Diagram: 1. P(AB) 2. P(BA) 3. P(ACB) 4. P(BCA) 5. P(ACBC)

Example Consider the experiment of rolling a fair die twice All outcomes in S are equally likely

Example, Using Table Let E=the sum of the faces is even Let S2=the second die is a 2 Find 1. P(S2E) 2. P(ES2)

Example, Using Table One way of doing this is to construct a table of frequencies: Event A Event Ac TOTALS Event B Total B Event Bc Total Bc Total A Total Ac Grand Total

Example, con’t Let’s try it to find P(S2E) and P(ES2) Event E Event Ec TOTALS Event S2 Event S2c Remember:

Sample Space for Rolling Two Die

Example 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

Example-Tree Diagram 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?

Independent Events If two events are independent, the occurrence of one event has no effect on the probability of the other. E and F are independent events if P(EF)=P(E)P(F) Similarly, P(EF G)=P(E)  P(F)  P(G), etc Independence of E and F implies that P(EF)=P(E) and P(F)= P(FE) If the events are not independent, then they are dependent.

Independent Events Consider flipping a coin and recording the outcome each time. Are these events independent? Let Hn=the event that a head comes up on the nth toss What is the P(H1 H2)? What is the probability P(H1 H2 H3)?

Independent Events You throw two fair die, one is green and the other is red, and observe the outcomes. Let A be the event that their sum is 7 Let B be the event that the red die shows an even # Are these events independent? Explain. Are these events mutually exclusive? Explain

Independent Events, con’t 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 5 B: the red die shows a 2 2. A: the sum is 5 B: the sum is less than 4 3. A: the sum is even B: the red die is even No, no, yes

Conditional Probability and Independence If E, F and G are three events, then E and F are independent, given that G has happened, if P(EFG)=P(EG)  P(FG) Likewise, events E, F and G are independent, given that H has happened, given that G has happened, if P(EFGH)=P(EH)  P(FH)  P(GH)

Independence 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?