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Critical Thinking All of the following examples came from Daniel Kahneman’s book Thinking, Fast and Slow. Question #1 Here is a simple puzzle. Do not try to solve it but listen to your intuition: A bat and ball cost $1.10. The bat costs one dollar more than the ball. How much does the ball cost? The ball costs $0.05. Question #2 Try to determine, as quickly as you can, if the argument is logically valid. Does the conclusion follow the premises? All roses are flowers. Some flowers fade quickly. Therefore some roses fade quickly. It is possible that there are no roses among the flowers that fade quickly. Question #3 If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets? 100 minutes or 5 minutes? 5 minutes Question #4 In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake? 24 days or 47 days? 47 days #5 Question 6 for group A How happy are you these days? How many dates did you have last month? On a scale of 1 to 10 with 10 being the happiest, how happy are you? Question 6 for group B How many dates did you have last month? How happy are you these days? On a scale of 1 to 10 with 10 being the happiest, how happy are you? Question #7 Take the sex of six babies born in sequence at a hospital. Consider three possible sequences: BBBGGG GGGGGG BGBBGB Which sequence is most likely? Because the events are independent and because the outcomes B and G are (approximately) equally likely, then any possible sequence of six births is as likely as any other. Question 8 for group A Anchoring Is the height of the tallest redwood more or less than 1,200 feet? What is your best guess about the height of the tallest redwood? Question 8 for group B Anchoring Is the height of the tallest redwood more or less than 180 feet? What is your best guess about the height of the tallest redwood? Question 9 Consider the pairs of causes of death. Indicate the more frequent cause and estimate the ratio of the two frequencies. Strokes and Accidents Asthma and Tornadoes Lightning and Botulism Disease and Accidents Diabetes and Accidents answer 9 Strokes cause almost twice as many deaths as all accidents combined. Asthma causes 20 times more deaths than tornadoes Death by lightning is 52 more frequent than by botulism Death by disease is 18 times more likely as accidental life Death by diabetes is 4 times more likely than by accident Question #10 Business administration Computer science Engineering Humanities and education Pre-Law Library science Physical and life sciences Social science & social work The following is a personality sketch of Tom W written during Tom’s senior year in high school by a psychologist, on the basis of psychological tests of uncertain validity: Tom W is of high intelligence, although lacking in true creativity. He has a need for order and clarity, and for neat and tidy systems in which every detail finds its appropriate place. His writing is rather dull and mechanical, occasionally enlivened by somewhat corny puns and flashes of imagination of the sci-fi type. He has a strong drive for competence. He seems to have little feel and little sympathy for other people, and does not enjoy interacting with others. Self-centered, he nonetheless has a deep moral sense. Now please take a piece of paper and rank the fields of specialization in order of the likelihood that Tom W is now a student in each of these fields. Tom W was intentionally designed as an “antibase- rate” character, a good fit to small groups and a poor fit to the most populated specialties. Substitution is perfect in this case: there is no indication that participants did anything else but judge representativeness. The question about probability (likelihood) was difficult, but the question about similarity was easier, and people answer it instead. Business administration Computer science Engineering Humanities and education Pre-Law Library science Physical and life sciences Social science & social work 17.3 % of students 2.8 5.3 24.8 4.9 0.9 13.2 8.6 Question #11 Linda is thirty-one years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations. Which statement is more likely correct Linda is a bank teller who is active in the feminist movement Linda is a bank teller Think in terms of Venn diagrams. The set of feminist bank tellers is wholly included in the set of bank tellers, as every feminist bank teller is a bank teller. Question #12 Price sets of dinnerware offered in a local store, where dinnerware regularly runs between $30 and $60. Assuming the dishes in the two sets are of equal quality. Please set a price for Set A and a price for Set B. Please set a price for Set A and a price for Set B. Question #13 A regular six-sided die has four green faces and to red faces will be rolled 20 times. Which sequence will be more likely? RGRRR GRGRRR GRRRRR Because the die has twice as many green sides as red sides, the first sequence is quite unrepresentative. The second sequence, which contains six tosses, is a better fit to what we expect from this die, because it includes two G’s. However, this sequence was constructed by adding a G to the beginning of the first sequence, so it can only be less likely than the first. Example 14 A cab was involved in a hit and run accident at night. Two companies, the Green and Blue, operate in the city. You are given the following data: Example 14 85% of the cabs in the city are Green and 15% Blue. A witness identified the cab as Blue. The court tested the reliability of the witness under the circumstances that existed on the night of the accident and concluded that the witness correctly identified each of the two colors 80% of the time and failed 20% of the time. What is the probability the cab involved in the accident was Blue? ′ Bayes Theorem |′ ′ = = ′| × ′ ′| × ′ | × + ′ | × .80×.15 .80×.15+ .20 ×.85 OR 12 12 +17 baseline Blue Green B G witness Blue B’ 12 17 Green G’ 3 68 15 85 reports 29 Overconfidence #15 For a number of years, professors at Duke conducted a survey in which CFO’s of large corporations estimated the returns of the S&P index over the following year. The Duke scholars collected 11,600 such forecasts and examined their accuracy. Overconfidence #15 The conclusion was straightforward: CFOs of larger corporations had no clue about the shortterm future of the stock market; the correlation between their estimates and the true value was slightly less than zero! The truly bad news is that the CFOs did not appear to know that their forecasts were worthless. Overconfidence #15 In addition to their estimates, participants provided two other estimates: a value that they were 90% sure would be too high, and one that they were 90% sure would be too low. The range between the two values is called an “80% confidence interval” and outcomes that fall outside the interval are labeled surprises. … Overconfidence #15 … As frequently happens in such exercises, there were far too many surprises; their incidence was 67%, more than 3 times higher than expected. This shows that CFOs were grossly overconfident about their ability to forecast the market. Overconfidence #15 The confidence interval that properly reflects the CFOs knowledge is more than four times wider than the intervals they actually stated. Gamble #16 Anthony’s current wealth is 1 million Betty’s current wealth is 4 million. They are both offered a choice between a gamble and a sure thing. The gamble: equal chances to end up owning 1 or 4 million OR The sure thing: own 2 million for sure Gamble #16 Anthony and Betty face the same outcomes: their expected wealth will be b 2.5 million if they take the gamble and 2 million if they prefer the sure-thing option. Gamble #16 Anthony (who currently owns 1 million): If Anthony chooses the sure thing, his wealth will double with certainty. This is very attractive. Alternatively, he can take a gamble with equal chances to quadruple his wealth or gain nothing. Gamble #16 Betty (who currently owns 4 million): If Betty chooses the sure thing, she loses half of her wealth with certainty, which is awful. Alternatively, she can take a gamble with equal chances to lose three quarters of her wealth or to lose nothing. Gamble #17 part 1 In addition to whatever you own, you have been given $1,000. You are now asked to choose one of these options: 50% chance to win $1,000 OR get $500 for sure Gamble #17 part 2 In addition to whatever you own, you have been given $2,000. You are now asked to choose one of these options: 50% chance to lose $1,000 OR lose $500 for sure Gamble #17 The outcomes are identical 50% $2,000 50% $1,000 Or $1,500 with certainty Allais’s Paradox #18 A: 61% chance to win $520,000 OR 63% chance to win $500,000 B: 98% chance to win $520,000 OR 100% chance to win $500,000 Imagine the outcome will be determined by a blind draw from an urn containing 100 marbles – you win if you draw a red marble, you lose if you draw white. A: 61% chance to win $520,000 OR 63% chance to win $500,000 B: 98% chance to win $520,000 OR 100% chance to win $500,000 In problem A, almost everyone prefers the left-hand urn, although it has fewer winning red marbles, because the difference in the size of the prize is more impressive than the difference in the chances of winning. In problem B, a large majority choose the urn with that guarantees a gain of $500,000. A: 61% chance to win $520,000 OR 63% chance to win $500,000 B: 98% chance to win $520,000 OR 100% chance to win $500,000 The two urns in problem B are more favorable versions of problem A, with 37 white marbles replaced by red winning marbles in each earn. The improvement of the left is clearly superior because each red marble gives you a chance to win $520,000 on the left and only $500,000 on the right. So you started in the first problem with a preference for the left-hand urn, which was improved more than the right-hand urn – but now you prefer the urn on the right. A: 61% chance to win $520,000 OR 63% chance to win $500,000 B: 98% chance to win $520,000 OR 100% chance to win $500,000 A: 61% chance to win $520,000 OR 63% chance to win $500,000 C: 37% chance to win $520,000 OR 37% chance to win $500,000 = 98% chance to win $520,000 OR 100% chance to win $500,000 NBA playoffs #19 When I wrote this the following teams had the 8 best records in the NBA Miami Heat San Antonio Spurs Oklahoma City Thunder LA Clippers Memphis Grizzlies Denver Nuggets Indiana Pacers Brooklyn Nets NBA playoffs Estimate the probability the following team will win the NBA playoffs? Miami Heat San Antonio Spurs Oklahoma City Thunder LA Clippers Memphis Grizzlies Denver Nuggets Indiana Pacers Brooklyn Nets Concurrent decisions #20 Imagine that you face the following pair of concurrent decision. First examine both decision, then make your choices. Concurrent decisions #20 Decision (i): Choose between Sure gain of $240 25% chance - gain $1,000 and 75% chance - gain nothing Decision (ii): Choose between Sure loss of $750 75% chance - lose $1,000 and 25% chance - lose nothing Concurrent decisions #20 combined AC: sure loss $510 AD: 25% win $240 and 75% lose $760 BC: 25% win $250 and 75% lose $750 BD 6.25% win $1,000, 37.5% $0, 56.25% lose $1,000 EV(AC) = loss $510 EV(BC) = loss $500 EV(AD) = loss $510 EV(BD) = loss $500 A repeated gamble #21 The great Paul Samuelson famously asked a friend whether he would accept a gamble on the toss of a coin in which he could lose $100 or win $200. Would you take such a gamble? Would you take a similar gamble if you were allowed to make 100 such bets? One Bet 50% chance you win $200 50% chance you lose $100 Two Bets 25% chance you win $400 50% chance you win $100 25% chance you lose $200 Three Bets 12.5% chance you win $600 37.5% chance you win $300 37.5% chance you break even 12.5% chance you lose $300 A repeated gamble #21 According to the book there is 1/2,300 chance of losing any money. Notice 34 heads and 66 tails results in a $200 gain And 33 heads and 67 tails results in a $100 loss Victim Compensation 22 (group A) You have the task of setting compensation for victims of violent crimes. You consider the case of a man who lost the use of his right arm as a result of a gunshot wound. He was shot when he walked in on a robbery occurring in a convenience store in his neighborhood. Victim Compensation 22 (group A) Two stores were located near the victim’s home, one of which he frequented more regularly than the other. The burglary happened in the man’s regular store. Victim Compensation 22 (group B) Two stores were located near the victim’s home, one of which he frequented more regularly than the other. The man’s regular store was closed for a funeral, so he did his shopping in the other store. Roulette Wheel 23 You are offered a choice between two bets, which are to be played on a roulette wheel with 36 sectors. Bet A: 11/36 to win $180, 25/36 to lose $15 Bet B: 35/36 to win $40, 1/36 to lose $10 Roulette Wheel 23 Bet A: 11/36 to win $180, 25/36 to lose $15 Bet B: 35/36 to win $40, 1/36 to lose $10 EV(A) = $44.58 EV(B) = $38.61 Epidemic 24 (group A) Imagine that the United States 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 estimates of the consequences of the programs are as follows: Epidemic 24 (group 1) If program A is adopted, 200 people will be saved. If program B is adopted, there is a one-third probability that 600 people will be saved and a two-thirds probability that no people will be saved. Epidemic 24 (group 2) If program A is adopted, 400 people will die. If program B is adopted, there is a one-third probability that nobody will die and a two-thirds probability that 600 people will die. Epidemic 24 The outcomes are exactly the same for Group 1 and Group2. They are simply framed differently. Theater Tickets #25 (group A) A woman has bought two $80 tickets to the theater. When she arrives at the theater, she opens her wallet and discovers that the tickets are missing. Will she but two more tickets to see the play? Theater Tickets #25 (group B) A woman goes to the theater, intending to buy two tickets that cost $80 each. She arrives at the theater, opens her wallet, and discovers to her dismay that the $160 with which she was going to make the purchase is missing. She could use her credit card. Will she buy the tickets? Gas Mileage #26 Consider two car owners who seek to reduce their costs: Adam switches from a gas-guzzler of 12 mpg to a slightly less voracious guzzler that runs at 14 mpg. The environmentally friendly Beth switches from a 30 mpg car to one that runs at 40 mpg. Suppose both drivers travel equal distances over a year. Who will save more gas by switching? Gas Mileage #26 If both car owners drive 10,000 miles, Adam will reduce his consumption from 833 gallons to 714 gallons, for a saving of 119 gallons. Beth’s use of fuel will drop from 333 gallons to 250, saving only 83 gallons. Gas Mileage #26 Decisions would improve if we used gallons per mile rather than mpg. Adam would reduce his usage by .0119 gpm ( .0833 gpm to .0714 gpm). Beth would reduce her usage by .0083 gpm (.0333 to .0250). This number can easily be multiplied by the number of miles driven. Experience #27 Each participant experienced two cold-hand episodes: Experience #27 The short episode consisted of 60 seconds of immersion in water at 14 degrees Celsius, which is experienced as painfully cold, but not intolerable. At the end of 60 seconds, the experimenter instructed the participant to remove his hand from the water and offered a warm towel. Experience #27 The long episode lasted 90 seconds. Its first 60 seconds were identical to the short episode. The experimenter said nothing at all at the end of the 60 seconds. Instead he opened a valve that allowed slightly warmer water to flow into the tub. During the additional 30 seconds, the temperature of the water rose by roughly 1 degree, just enough for most subjects to detect a slight decrease in the intensity of the pain. Experience #27 Participants were told that they would have 3 cold-hand trials, but in fact they experienced only 2 episodes. Trials were separated by 7 minutes. 7 minutes after the 2nd trial, the participants were given a choice about the 3rd trial. They were told that one of the experiences would be repeated exactly, and they were free to choose whether to repeat the short episode or the long episode. Experience #27 Fully 80% of the participants reported that their pain diminished during the final phase of the longer episode opted to repeat it. Thereby declaring themselves willing to suffer 30 seconds of needless pain in the anticipated third trial. Final Thoughts Nothing in life is as important as you think it is when you are thinking about it. Final Thoughts How much pleasure do you get from your car” When do you get pleasure from your car? You get pleasure or displeasure from your car when you think about your car, which is probably not very often. You probably answered the question “How much pleasure do you get from your car when you think about it I need to acknowledge and thank Daniel Kahneman. All of these examples came from his book Thinking, Fast and Slow.