Oct. 21 Assignment: Read Chapter 17 Try exercises 5, 13, and 18 on pp. 379 380 Rank the following causes of death in the US from most common to least common: Stroke Homicide Your answers may depend on the Availability Heuristic Alzheimer's Suicide
Quick review Suppose that you roll a fair 6-sided die until the first occurrence of a 4. What is the probability that the first 4 occurs on the third roll? (A) 1/6 (B) 5/6 (C) (5/6) (5/6) (1/6) (D) (1/6) (1/6) (1/6) (E) (1/6) + (1/6) + (1/6)
Example of Kahneman and Tversky s work Imagine that the U.S. is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume that the exact scientific estimate of the consequences of the programs are as follows: (A) If Program A is adopted, 200 people will be saved. (B) If Program B is adopted, there is a 1/3 probability that 600 people will be saved and a 2/3 probability that no people will be saved. Which of the two programs would you favor, A or B?
Mary likes earrings and spends time at festivals shopping for jewelry. Her boy friend and several of her close girl friends have tattoos. They have encouraged her to also get a tattoo. Unknown to you, Mary will be sitting next to you in the next STAT 100 class. Rank the following statements from most likely to least likely: A. Mary is a physics major. B. Mary is a physics major with pierced ears. C. Mary has pierced ears.
Do you remember... Rule 4: If the occurrence of one event forces the occurrence of another event, then the probability of the second event is always at least as large as the probability of the first event. If event A forces B to occur, then P(A) < P(B) Special case: P(E and F) < P(E) P(E and F) < P(F)
Putting B (Mary is a physics major with pierced ears) ANYWHERE BUT LAST is impossible and illustrates the Conjunction fallacy: assigning higher probability to a detailed scenario involving the conjunction of events than to one of the simple events that make up the conjunction. A possible cause of this fallacy is the Representative heuristic: leads people to assign higher probabilities than are warranted to scenarios that are representative of how we imagine things would happen.
Kahneman and Tversky example, page 353: If a population consists of 30 engineers and 70 lawyers, what is the probability that a randomly selected individual is an engineer? Daniel Kahneman Amos Tversky Next question: Dan is a 30-year-old man. He is married with no children. A man of high ability and high motivation, he promises to be quite successful in his field. He is well liked by his colleagues. What is the probability that Dan is an engineer?
Forgotten base rates Another example: Suppose that a particular disease affects 1% of those who get tested for it. Also suppose that the test is 98% accurate. What would you advise a patient who tests positive if the test result were the only piece of information? True probability of disease: about 33%
Calibrating personal probabilities of experts: p. 357
Another Example of Kahneman and Tversky s work Imagine that the U.S. is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume that the exact scientific estimate of the consequences of the programs are as follows: c) If Program C is adopted, 400 people will die. d) If Program D is adopted, there is a 1/3 probability that nobody will die and a 2/3 probability that 600 people will die. Which of the two programs would you favor, C or D?
p. 359, exercise 5 A defense attorney is trying to convince the jury that his client s wallet was planted at the scene of the crime by the gardener. Here are two possible statements: (A) The gardener dropped the wallet when no one was looking. (B) The gardener dropped the wallet in his sock and when no one was looking he quickly reached down and lowered his sock, allowing to wallet to drop out. Statement (B) cannot have a higher probability than statement (A). Yet jurors may assign it a higher probability anyway. Why?
p. 359, exercise 7 A telephone solicitor asking for money for a charity says, We are asking for as much as you can give, up to $300. Do you think the results would be different if the solicitor said, We typically get $25 to $50, but give as much as you can? Why?
p. 360, exercise 14c If a word is chosen at random from, say, a newspaper article, which is more likely? (A) It begins with the letter k (B) It has k as the third letter K&T have shown that people tend to answer (A), even though there are about twice as many words with k in the third position. Why?
Some main ideas from Chpt. 16 n Representativeness heuristic (and conjunction fallacy) n Anchoring n Availability heuristic n Forgotten base rates Can you recall an example for each of these?
Other ideas from Chpt. 16 (pp. 354 355) n People tend to be optimistic n People tend to be conservative (in the literal sense) and overconfident
Other ideas from Chpt. 16, leading into Chpt. 17 (pp. 346 347) n People prefer certainty to near-certainty, even if it results in a decreased expected gain; this is reversed for negative outcomes. n People prefer a small possibility of a large gain to none, even if it results in a decreased expected gain; this is reversed for negative outcomes. n People would rather eliminate some negative possibilities than reduce all negative possibilities, even if the expected gain is the same.
Should people always use expectations (expected values) to make decisions? Given the choice, which of the following options would you prefer? (A) A gift of $240, guaranteed (B) A 25% chance to win $1000 (but nothing if you lose)
Should people always use expectations (expected values) to make decisions? Given the choice, which of the following options would you prefer? (A) A loss of $740, guaranteed (B) A 75% chance to lose $1000 (no loss if you win)
Should people always use expectations (expected values) to make decisions? How much would you pay to play the following game? Flip a coin. If you get heads on flip #1, you win $1. Otherwise, keep flipping. If you get heads on flip #2, you win $2. Otherwise, keep flipping. If you get heads on flip #3, you win $4. Otherwise, keep flipping. If you get heads on flip #4, you win $8. Otherwise, keep flipping. And so on Basically, your potential winnings double each flip, but the game ends whenever the first heads appears.
Should people always use expectations (expected values) to make decisions? Based on evidence of the previous example, the answer is NO. (The previous example is called the St. Petersburg paradox.) NB: Section 17.5 seems to imply that the answer is YES. I only sometimes agree with this.
Forgotten base rates revisited Suppose that a particular disease affects 1% of those who get tested for it. Also suppose that the test is 98% accurate. What would you advise a patient who tests positive if the test result were the only piece of information? True probability of disease: about 33% How would we find this probability?
Cancer testing: confusion of the inverse Suppose we have a cancer test for a certain type of cancer. Sensitivity of the test: If you have cancer then the probability of a positive test is.98. Pr(+ given you have C) =.98 Specificity of the test: If you do not have cancer then the probability of a negative test is.98. Pr(- given you do not have C) =.98 Base rate: The percent of the population who has the cancer. This is the probability that someone has C. Suppose for our example it is 1%. Hence, Pr(C) =.01.
Percent table + Positive - Negative Sensitivity C (Cancer).98.02.01 Base Rate Specificity no C (no Cancer).02.98.99 false positive false negative Suppose you go in for a test and it comes back positive. What is the probability that you have cancer?
Table of proportions (given): + - Base rate C.98.02.01 no C.02.98.99 Hypothetical table of counts: + - C 98 2 100 no C 198 9702 9,900 296 9704 10,000 Pr(C given a positive test result) = 98/296 = 33.1%