Sta220 - Statistics Mr. Smith Room 310 Class #9

Section

Problem: To develop programs for business travelers staying at convention hotels, Hyatt Hotels Corp. commissioned a study on executives who play golf. The study revealed that 55% of the executives admitted that they had cheated at golf. Also, 20% of the executives admitted that they had cheated at golf and had lied in business. Given that an executive had cheated at golf, what is the probability that they executive also had lied in business?

The event probabilities we’ve been discussing give the relative frequencies of the occurrences of the events when the experiment is repeated a very large number of times. Such probabilities are often called unconditional probabilities.

3.5 Conditional Probability A probability that reflects such additional knowledge is called the conditional probability of the event.

Example: We’ve seen that the probability of an even number (event A) on a toss of a fair die is ½. But suppose we’re given the information that on a particular throw of the die the results was a number less than or equal to 3 (event B). Would the probability of observing an even number on that throw of the die still be equal to ½?

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Reduced sample space for the die-toss experiment: given that event B has occurred

Problem: To develop programs for business travelers staying at convention hotels, Hyatt Hotels Corp. commissioned a study on executives who play golf. The study revealed that 55% of the executives admitted that they had cheated at golf. Also, 20% of the executives admitted that they had cheated at golf and had lied in business. Given that an executive had cheated at golf, what is the probability that they executive also had lied in business?

Solution

Thus, given that a certain executive had cheated at golf, the probability that the executive also had lied in business is.364.

Problem:

Solution

The conditional probability that an adult male smoker develops cancer (.20) is five times the probability that a nonsmoker develop cancer (.04). This relationship does not imply that smoking causes cancer, but it does suggest a pronounced link between smoking and cancer.

3.6 The Multiplicative Rule and Independent Events The probability of an intersection of two events can be calculated with the multiplicative rule.

Problem: In a classic psychology study conducted in the early 1960s, Stanley Milgram performed a series of experiment in which a teacher is asked to shock a learner who is attempting to memorize word pairs whenever the learner gives the wrong answer. The shock levels increase with each successive wrong answer. {Unknown to the teacher, the shocks are not real.} Two events of interest are A: {The teacher “applies” a sever shock (450 volts).} B: {The learner protest verbally prior to receiving the shock.} A recent application of Milgram’s shock study revealed that P(B) =.5 and P(A|B) =.7. On the basis of the information, what is the probability that a learner will protest verbally and a teacher will apply a severe shock?

Solution

Problem: A county welfare agency employs 10 welfare workers who interview prospective food stamp recipients. Periodically, the supervisor selects, at random, the forms completed by two workers and subsequently audits them for illegal deductions. Unknown to the supervisor, three of the workers have regularly been giving illegal deductions to applicants. What is the probability that both of the workers chosen have been giving illegal deductions?

Solution Let N: Non-Illegal I: Illegal

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Tree diagram

We have showed that the probability of event A may be substantially altered by the knowledge that an event B has occurred. However, this will not always be the case; in some instances, the assumption that event B has occurred will not alter the probability of event A at all.

Problem: Consider the experiment of tossing a fair die, and let A = {Observe an even number.} B = {Observe a number less than or equal to 4.} Are A and B independent events?

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Venn diagram for die-toss experiment

Solution

Thus, assuming that the occurrence of event B does not alter the probability of observing an even number, the probability remains ½. Therefore, the events A and B are independent.

Problem

Solution