Bayes’ Rule

A college senior was asked this question in an interview for a banking job. A can contains 20 coins, 19 normal coins and 1 coin that has heads on both sides. A coin is randomly selected from the can and flipped five times. It lands heads all five times. What is the probability that it is the two-headed coin?

The landmark 1964 US Surgeon General report on cigarette smoking cited these data: $6 \%$ of all deaths are due to lung cancer; $85 \%$ of the people who die of lung cancer are smokers; and 1/3 of the adult population are smokers. Compare the probabilities of dying of lung cancer for smokers and nonsmokers.

Suppose that $10 \%$ of student essays are written by ChatGPT and that a ChatGPT detection program is $90 \%$ accurate: it correctly flags $90 \%$ of ChatGPT-written essays as having been written by ChatGPT and it correctly flags $90 \%$ of human-written essays as having been written by humans. What percent of the essays flagged as written by ChatGPT were in fact written by humans?

Answer this question that doctors and residents at a Boston hospital were asked: one in a thousand people have a certain disease. A test for the disease always gives a positive result if a person has the disease and has a $5 \%$ chance of giving a false positive result for a patient who does not have the disease. What percentage of the people with positive test results actually have the disease?

In a mock courtroom experiment, 144 volunteer jurors were told that a liquor store had been robbed by a man wearing a ski mask. The police arrested a suspect near the store whose height, weight, and clothing matched the clerk's description. The ski mask and money were found in a nearby trash can. After hearing this evidence, the jurors were asked to write down their estimate of the probability that the arrested man "really did it." The average probability was .25.

Then a forensic expert testified that samples of the suspect's hair matched a hair found inside the ski mask and that only $2 \%$ of the population has hair that matches the hair found in the ski mask. After hearing this evidence, the jurors' average revised probability of the defendant's guilt was .63. What is the Bayesian posterior probability?

In 2016, two Chinese researchers reported that a computer analysis of the faces of male Chinese criminals and non-criminals correctly identified $89.5 \%$ of the criminals as criminals and correctly identified $93 \%$ of the non-criminals as non-criminals. If $.36 \%$ of the Chinese male population is criminal, what is the probability that a person labeled criminal by the computer is actually a criminal?

Some cars turn out to be "lemons," with many defects. Unfortunately, the quality of a new car cannot be determined until the car has been driven several hundred miles. The unlucky purchaser of a lemon may try to unload it, by resale to someone else. Suppose that $10 \%$ of all new cars are lemons and that $90 \%$ of all lemons and $5 \%$ of all non-lemons are offered for sale within the first year of ownership. If so, what is the probability that a randomly selected car from among those offered for sale within the first year is a lemon?

A letter to Ask Marilyn began: "As professors of statistics, we found your response to the drug-testing question perplexing and, indeed, incorrect." The question was,

Suppose we assume that $5 \%$ of people are drug-users. A test is $95 \%$ accurate, which we'll say means that if a person is a user, the result is positive $95 \%$ of the time; and if she or he isn't, it's negative $95 \%$ of the time. A randomly selected person tests positive. Is the individual highly likely to be a drug user?

In fact, Marilyn calculated the probability correctly. What is the correct probability?

To see who serves first in their Thursday squash games, Player A spins the racket while Player B guesses whether the racket logo will stop face up or face down. Player B initially believes that there is only a $5 \%$ chance that A cheats when he spins the racket. But after A wins the first five times he spins the racket, B isn't so sure. Assuming that A will always win if he cheats and has a $50 \%$ chance of winning if he doesn't cheat, what is B's revised probability that A cheats?

In three careful studies, polygraph experts tested several persons, some known to be truthful and the others known to be lying, to see if the experts could tell which were which. Overall, $83 \%$ of the liars were pronounced "deceptive" and $57 \%$ of the truthful people were judged "honest." Using these data and assuming that $80 \%$ of the people tested are truthful and $20 \%$ are lying, what is the probability that a person pronounced "deceptive" is in fact truthful? What is the probability that a person judged "honest" is lying?

A magician claims she has practiced coin flips millions of times and can flip heads $75 \%$ of the time. She shows you a coin which appears to be a normal coin before she begins her demonstration. You think there are three possibilities:

Approximately $1.5 \%$ of Americans are schizophrenic. Computed axial tomography (CAT) scans show brain atrophy in $30 \%$ of people who have been diagnosed as schizophrenic and in only $2 \%$ of people diagnosed as not schizophrenic. In the 1982 trial of John Hinckley for the attempted assassination of President Ronald Reagan, the defense attorney tried to present evidence that a CAT scan of Hinckley had shown brain atrophy, thereby indicating that Hinckley was schizophrenic. Is a CAT scan showing brain atrophy persuasive evidence that a person is schizophrenic?

Epic claimed that sepsis predictions made by its AI Epic Sepsis Model (ESM) are $76 \% -83 \%$ accurate, but there were no independent tests until a medical team examined the hospital records of 38,455 patients at Michigan Medicine, of whom 2,552 (6.6) experienced sepsis. ESM generated a (correct) sepsis alert for 843 of these 2,552 patients and (correctly) did not generate a sepsis alert for 29,775 of the 35,903 patients who did not have sepsis. What fraction of ESM's sepsis alerts was false positive?

For a woman who gives birth at age 35, the probability of having a baby suffering from Down syndrome is $1 / 270$. A test of the amniotic fluid in the mother's uterus is virtually $100 \%$ accurate in predicting Down syndrome, but is expensive and can cause a miscarriage. A study of the effectiveness of an inexpensive blood test that does not risk miscarriage found that in $89 \%$ of the Down-syndrome cases, the test gave a positive reading, while in $75 \%$ of the cases without Down syndrome the test gave a negative reading. Of those cases where there is a positive reading, what fraction is false positive?

Tay-Sachs disease is a genetic disorder that occurs in roughly 1 in 320,000 US babies. Suppose that a genetic test is $99 \%$ accurate (a positive result) for babies that have Tay-Sachs and $98 \%$ accurate (a negative result) for babies that do not have Tay-Sachs. If a test comes back positive, what is the probability the baby has Tay-Sachs?

A new theory groups people into one of three personality types ($\mathrm{X}, \mathrm{Y}$, or Z) and assumes that one-third of the population is in each group. A large empirical study asked people this question, "Would you rather go to the beach or the mountains for a two-week vacation?," and found that $70 \%$ of the Type X people answered "Beach," as did $50 \%$ of the Type Y people, and $30 \%$ of the Type Z people.

If Jill Jones answers "beach," what is the probability that she is a Type X personality?

"Five out of ten people who have this disease die from it. It is lucky you came to me; I've had five patients and they all died." Suppose that there are three kinds of doctors: excellent $(10 \% )$, good $(80 \% )$, and incompetent $(10 \% )$. With an excellent doctor, every patient lives; with a good doctor, half live and half die; and with an incompetent doctor, every patient dies. Given this doctor's track record (five of five died), what is the probability that he/she is incompetent?

Suppose that $20 \%$ of the people who are given breathalyzer tests for intoxication are legally intoxicated and $80 \%$ are not, and that the test gives a positive reading $85 \%$ of the time when a person is intoxicated, and gives a negative reading $80 \%$ of the time when a person is not intoxicated. If the test comes back positive, what is the probability that the person who was tested is, in fact, not intoxicated?

Suppose that $10 \%$ of all investment advisers are experts and $90 \%$ are guessers. Each expert has a .6 probability of making a correct prediction whether the stock market will beat the bond market over the course of a year; a guesser has a .5 probability of being correct. Each adviser's chances of being correct are independent of the other advisers and also independent of whether they were correct in earlier years. If an adviser is correct 5 years in a row, what is the probability that this adviser is an expert?

A 1960s study found that in 3,379 criminal jury trials where the jury reached a verdict, the judge agreed with the jury's verdict in 2,696 cases (79.8)-with both judge and jury convicting in 2,217 cases (65.6) and both acquitting in 479 cases $(14.2 \% )$. In 604 cases $(17.9 \% )$, the jury acquitted but the judge would have convicted; in 79 cases $(2.3 \% )$, the jury convicted but the judge would have acquitted.

Israel has a high COVID-19 vaccination rate, yet on August 15, 2021, $58 \%$ of the Israelis hospitalized for COVID-19 were fully vacci-nated-suggesting that vaccinations are ineffective or even harmful. Why is this $58 \%$ number misleading from a purely statistical standpoint? What statistical technique would you use to solve this problem?

Here's a stylized example of how polls can be used to predict election outcomes. Suppose that there are only two possibilities: Cameron is preferred by either $51 \%$ or $49 \%$ of all voters, and our prior probability is that each situation is equally likely. A poll of 1,000 voters finds that $510(51 \% )$ of those surveyed prefer Cameron. What is the posterior probability that $51 \%$ of all voters prefer Cameron?

Suppose that $90 \%$ of the people who enjoyed the television show Homicide: Life on the Street also enjoyed the show The Wire, while $20 \%$ of the people who did not enjoy Homicide did enjoy The Wire. If $40 \%$ of all people enjoyed Homicide, what is the probability that a randomly selected person who enjoyed The Wire also enjoyed Homicide?

Fifteen percent of US households live below the poverty line. In a third of all US households, a woman is the sole income provider. In $60 \%$ of poor households, a woman is the sole income provider. Of those households in which a woman is the sole income provider, what fraction is poor?

A farmer sends out 10,000 cartons of eggs every day. Two inspectors independently glance at each carton, looking for cracked eggs, and each inspector has a .70 probability of noticing when a carton has cracked eggs. They never reject a carton that has no cracked eggs. What is the probability that a carton with cracked eggs will be noticed by at least one inspector? If $30 \%$ of the inspected cartons have cracked eggs, what is the probability that a carton that is passed by both inspector has cracked eggs?

It has been argued that diagonal earlobe creases for adult males indicate the presence of coronary heart disease. It has been estimated that $5 \%$ of all adult males have coronary heart disease and that $47 \%$ of those with coronary heart disease have diagonal earlobe creases, while $30 \%$ of those with no heart disease have earlobe creases. What is the probability that an adult male with an earlobe crease has heart disease?

In the US criminal defendants are presumed innocent until they are proven guilty beyond a reasonable doubt because it is thought "better to let nine guilty people go free than to send one innocent person to prison." Assume that $90 \%$ of all defendants are guilty, $90 \%$ of the guilty defendants are convicted, and $90 \%$ of the innocent defendants are set free. What percent of the people convicted are innocent? Of those people set free, what percent are guilty?

In Craig v. Boren (1976), the US Supreme Court considered whether important government objectives were served by the sex distinction in an Oklahoma statute that prohibited the sale of $3.2 \%$ beer to males under the age of 21 and to females under the age of 18. Among the evidence considered in this case were the following data on persons arrested in Oklahoma for driving under the influence (DUI) during the last four months of 1973: 92 of those arrested were male; $8 \%$ of the males arrested were under the age of 21 ; and $5 \%$ of the females arrested were under the age of 21. Assume a hypothetical population of 10,000 DUI arrests and construct a contingency table with the columns showing sex (male or female) and the rows showing age (under 21 or older). Do these data indicate that age and sex are strongly related or largely unrelated?

The Oakland Athletics, winners of 103 games during the regular season, were heavily favored to defeat the Cincinnati Reds in the 1990 World Series. After the Reds won the series in four straight games, several Athletics said that the series had been a fluke and that they were still convinced that they were the best team in baseball. One speculated that if they played 100,000 games against the Reds, the Athletics would win 60,000.

Assume that baseball games are Bernoulli trials with the Athletics having a probability p of winning a game against the Reds and that there are two possible values for p : .60 and .50. Before the 1990 World Series, the prior probabilities are $\mathrm{P}[\mathrm{p}=.60]=.70$ and $\mathrm{P}[\mathrm{p}=.50]=.30$. Use Bayes' theorem to determine the revised probability that $\mathrm{p}=.60$ after the Athletics lose four of four games to the Reds.

A routine examination discovers a lump in a female patient's breast. Only 1 of 100 such lumps turns out to be malignant, but, to be safe, the doctor orders a mammogram X-ray test. If the lump is malignant, there is a .8 probability that the mammogram reading will be positive; if the lump is benign, there is a .9 probability that the reading will be negative. The test comes back positive and the doctor orders a second test, which comes back negative. Assuming the test results to be independent, what is your estimate of the probability that this lump is malignant?