Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. Probability part 2 Chapter 7

Main rules for working with probabilities: 1.P(A c ) = 1-P(A) 2.(a). P(A or B) = P(A) + P(B) – P(A and B) (b). P(A or B) = P(A) + P(B) 3. (a). P(A and B) = P(A)P(B|A) (b). P(A and B) = P(A)P(B) (c). P(A and B and C) = P(A)P(B)P(C)… 4. P(B|A) = P(B and A)/P(A) 2

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc Strategies for Finding Complicated Probabilities Example 7.2 Winning the Lottery Event A = winning number is 956. What is P(A)? Method 1: With physical assumption that all 1000 possibilities are equally likely, P(A) = 1/1000. Method 2: Define three events, B 1 = 1 st digit is 9, B 2 = 2 nd digit is 5, B 3 = 3 rd digit is 6 Event A occurs if and only if all 3 of these events occur. Note: P(B 1 ) = P(B 2 ) = P(B 3 ) = 1/10. Since these events are all independent, we have P(A) = (1/10) 3 = 1/1000. * Can be more than one way to find a probability.

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 4 P(A and B): define event in physical terms and see if know probability. Else try multiplication rule (Rule 3). Series of independent events all happen: multiply all individual probabilities (Extension of Rule 3b) One of a collection of mutually exclusive events happens: add all individual probabilities (Rule 2b extended). Check if probability of complement easier, then subtract it from 1 (applying Rule 1). Hints and Advice for Finding Probabilities

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 5 None of a collection of mutually exclusive events happens: find probability one happens, then subtract that from 1. Conditional probability: define event in physical terms and see if know probability. Else try Rule 4 or next bullet as well. Know P(B|A) but want P(A|B): use tree diagrams! Hints and Advice for Finding Probabilities

6 Step 1: (a). List each separate random circumstance involved in the problem. (b). List the possible outcomes for each random circumstance. (c). Assign whatever probabilities you can with the knowledge you have. Step 2: Write the event for which you want to determine the probability in terms of the outcomes in Step 1. Pay attention to words like not, and, or, given. Step 3: Based on the key words in step 2 determine which probability rules can be combined to find the probability of interest. Look for any combination of ‘independent’, ‘disjoint’, ‘conditional probability’… Steps for Finding Probabilities (my version:)

Example: Assume that the probability that each birth in a family is a boy is and that the outcomes of successive births are independent. If the family has three children, what is the probability that they have two boys and one girl (in any order)? 7

Example: A particular brand of cereal box contains a prize in each box. There are four possible prizes, and any box is equally likely to contain each of the four prizes. What is the probability that you will receive two different prizes if you purchase two boxes? Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 8

9 Tree Diagrams Step 1: Determine first random circumstance in sequence, and create first set of branches for possible outcomes. Create one branch for each outcome, write probability on branch. Step 2: Determine next random circumstance and append branches for possible outcomes to each branch in step 1. Write associated conditional probabilities on branches. Step 3: Continue this process for as many steps as necessary. Step 4: To determine the probability of following any particular sequence of branches, multiply the probabilities on those branches. This is an application of Rule 3a. Step 5: To determine the probability of any collection of sequences of branches, add the individual probabilities for those sequences, as found in step 4. This is an application of Rule 2b.

Example: In a 1998 survey of most of the 9 th grade students in Minnesota, 22.9% of boys and 4.5% of girls admitted that they gambled at least once a week during the previous year. The population consisted of 50.9% girls and 49.1% boys. (a). What is the probability that a randomly selected student gambles weekly? (a). What is the probability that a randomly selected student will be a male who also gambles at least weekly? (b). Suppose that you select a student and find that the student is a gambler. What is the probability that the student is a girl? 10

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 11 Example (con’t): the hypothetical two-way table Sample of 9 th grade teens: 49.1% boys, 50.9% girls. Results: 22.9% of boys and 4.5% of girls admitted they gambled at least once a week during previous year. Start with hypothetical 100,000 teens … (.491)(100,000) = 49,100 boys and thus 50,900 girls Of the 49,100 boys, (.229)(49,100) = 11,244 would be weekly gamblers. Of the 50,900 girls, (.045)(50,900) = 2,291 would be weekly gamblers.

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 12 Example 7.8 Teens and Gambling (cont) P(boy and gambler) = P(girl | gambler) = P(gambler) =

13 Example: Last week, Alicia went to her physician for a routine medical exam. This morning her physician phoned to tell her that one of her tests came back positive, indicating that she may have a disease D. The physician told her that the the test is 95% accurate whether someone has disease D or not. In other words, when someone has disease D, the test detects it 95% of the time. When someone does not have D, the test is rightly negative 95% of the time. Therefore, according to the physician, even though only 1 in 1000 women of Alicia’s age actually has D, the test is a pretty good indicator that Alicia actually has disease D. Given the positive test result, what is the probability that Alicia has disease D?

Medical testing: The sensitivity of a test is the proportion of people who correctly test positive when they actually have the disease. The specificity of a test is the proportion of people who correctly test negative when they don’t have a disease. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 14

15 Example (con’t): ROUTINE MEDICAL TESTING VS. POPULATION AT RISK KEY QUESTIONS: What is the baseline risk in the population? What is the population?

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc Using Simulation to Estimate Probabilities Some probabilities so difficult or time- consuming to calculate – easier to simulate. If you simulate the random circumstance n times and the outcome of interest occurs in x out of those n times, then the estimated probability for the outcome of interest is x/n.

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 17 Example 7.19 Getting All the Prizes Cereal boxes each contain one of four prizes. Any box is equally likely to contain each of the four prizes. If buy 6 boxes, what is the probability you get all 4 prizes? Shown above are 50 simulations of generating a set of 6 digits, each equally likely to be 1, 2, 3, or 4. There are 19 bold outcomes in which all 4 prizes were collected. The estimated probability is 19/50 =.38. (Actual probability is.3809.)

Coincidences & Intuitive Judgments about Probability 1.Confusion of the Inverse 2.Specific People vs. Random Individuals 3.Coincidences 4.The Gambler’s Fallacy

19 Confusion of the Inverse Example: Diagnostic Testing Confuse the conditional probability “have the disease” given “a positive test result” -- P(Disease | Positive), with the conditional probability of “a positive test result” given “have the disease” -- P(Positive | Disease), also known as the sensitivity of the test. Often forget to incorporate the base rate for a disease.

A study of 100 physicians: One of your patients has a lump in her breast. You are almost certain that it is benign, in fact you would say there is only 1% chance that it is malignant. But just to be sure, you have the patient undergo a mammogram. You know from the medical literature that mammograms are 80% accurate for malignant lumps and 90% accurate for benign lumps. Sadly, the mammogram for your patient is returned with the news that the lump is malignant. What are the chances that it is truly malignant? 20

Most of the physicians to whom this question was posed answered that the probability was truly malignant was about 75%. What is the actual probability? What is the baseline risk in the population in this example? 21

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 22 Specific People versus Random Individuals In long run, about 50% of marriages end in divorce. At the beginning of a randomly selected marriage, the probability it will end in divorce is about.50. Does this statement apply to you personally? If you have had a terrific marriage for 30 years, your probability of ending in divorce is surely less than 50%. The chance that your marriage will end in divorce is 50%. Two correct ways to express the aggregate divorce statistics:

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 23 Coincidences Example 7.23 Winning the Lottery Twice A coincidence is a surprising concurrence of events, perceived as meaningfully related, with no apparent causal connection. In 1986, Ms. Adams won the NJ lottery twice in a short time period. NYT claimed odds of one person winning the top prize twice were about 1 in 17 trillion. Then in 1988, Mr. Humphries won the PA lottery twice. 1 in 17 trillion = probability that a specific individual who plays the lottery exactly twice will win both times. Millions of people play the lottery. It is not surprising that someone, somewhere, someday would win twice.

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 24 The Gambler’s Fallacy Primarily applies to independent events. Independent chance events have no memory. Example: Making ten bad gambles in a row doesn’t change the probability that the next gamble will also be bad. The gambler’s fallacy is the misperception of applying a long-run frequency in the short-run.