Why a 1 Per Cent Chance Feels Bigger
In the best-known estimates a 1 per cent chance carries a decision weight of about 5.5. The weight shifts with how vivid the outcome is, and in some experiments it reverses when people learn the odds by sampling.
Answer to the question
Why does a 1 per cent chance feel bigger?
A 1 per cent chance feels bigger because the mind gives a possible outcome more weight than its odds warrant. In Tversky and Kahneman's 1992 estimates it carries a decision weight of about 5.5. The weight rises for vivid outcomes, and in experiments where people learned the odds by sampling, rare events were underweighted instead.
Key takeaways
- In Amos Tversky and Daniel Kahneman's 1992 estimates, a 1 per cent chance of a gain carries a decision weight of about 5.5, and a 50 per cent chance carries 42.1.
- Kahneman writes that below 1 per cent no unique decision weight can be assigned to rare events, and that some are ignored entirely.
- Yuval Rottenstreich and Christopher Hsee found in 2001 that a prize low in emotion was preferred to one high in emotion when each was certain, and that the preference reversed at low probability.
- Ralph Hertwig and colleagues found in 2004 that choices from experience differed from choices from description by an average of 36 percentage points, and a 2006 reanalysis traced much of the gap to sampling error and a 2013 study found underweighting persisted without it.
Daniel Kahneman and Amos Tversky argued that people give a small chance more weight than its odds deserve, and then spent years measuring how much more.
Their measurement became a standard table of decision weights. In Tversky and Kahneman's 1992 estimates a 1 per cent chance of a gain carries a decision weight of about 5.5, more than five times its probability.
Allais and the last two points
Kahneman and Tversky published prospect theory in 1979, and the chapter of Thinking, Fast and Slow that sets out its weights begins with a demonstration by Maurice Allais. He put two problems, each a choice between two gambles, to some of the leading decision theorists in the world, and then revealed that their preferences violated the expected utility theory they worked with. Kahneman reports that Allais meant the announcement as a bombshell. In the first problem a 61 per cent chance of $520,000 stood against a 63 per cent chance of $500,000. In the second a 98 per cent chance of $520,000 stood against a certain $500,000.
The same two points of probability separate the options in both. Kahneman and Tversky's reading is that the points are not worth the same, because the step from 98 per cent to certainty is far more impressive than the step from 61 to 63. The book names this the certainty effect: outcomes that are almost certain get less weight than their probability justifies.
At the other end sits the possibility effect, in which unlikely outcomes get more weight than they deserve. A 1 per cent chance is a possibility, so it lands on the inflated side of the first step, which is, in Kahneman's words for the move from no chance to a 5 per cent chance, one "creating a possibility that did not exist earlier."
Where the number 5.5 comes from
The paper in the Journal of Risk and Uncertainty called the revised theory cumulative prospect theory and gave its weights a formula with one parameter for gains and another for losses. The median estimates were 0.61 for gains and 0.69 for losses, drawn from an experiment with 25 graduate students from Berkeley and Stanford choosing among monetary prospects. We ran the gain formula at 0.61. It reproduces every value that the text of Kahneman's book quotes from its table of decision weights, which names no paper beside it: 8.1, 13.2, 18.6, 71.2, 79.3 and 87.1.
The rows that matter here are below.
Very small probabilities can be either greatly overweighted or neglected altogether.
| Chance of the gain (%) | Decision weight |
|---|---|
| 1 | 5.5 |
| 5 | 13.2 |
| 10 | 18.6 |
| 50 | 42.1 |
| 90 | 71.2 |
| 95 | 79.3 |
| 99 | 91.2 |
Source: the 1992 gain formula at 0.61, calculated for this piece.
A 1 per cent chance is weighted at 5.5, a factor of five and a half. A 10 per cent chance is weighted at 18.6, under twice its probability, and a 50 per cent chance is underweighted at 42.1. From 5 to 95 per cent the weights run only from 13.2 to 79.3, which Kahneman calls "about two-thirds as much as rationally expected." The paper states the pattern in a sentence: people overweight low probabilities and underweight moderate and high ones, and "very small probabilities can be either greatly overweighted or neglected altogether."
The second half of that sentence bears on a 1 per cent chance. The 5.5 was estimated from choices involving gains, and its row sits at the edge of the range where Kahneman says the curve stops being dependable. He writes, just after the table, that below 1 per cent no unique decision weight can be assigned to rare events and that some are ignored entirely. The paper reports the weighting curves for gains and for losses as quite close, the former slightly more curved.
Does feeling change the weight?
A 2002 study of 226 experienced clinical psychologists took up a 2000 study by Paul Slovic, John Monahan and Donald MacGregor, whose abstract reports that describing dangerousness as a frequency made risk seem much greater than an equivalent probability did.
The 2002 authors reported the effect as clinicians being more likely to keep a patient hospitalised when risk was framed as a frequency, 20 out of 100 patients, than as a probability, 20 per cent, and attributed it to frequencies being easier to visualise. The 2002 study found that frequency formatting and a vivid depiction of the violent outcome both produced more conservative decisions, but only among clinicians who worked in forensic facilities. The abstract's example is a 20 per cent chance, so these studies do not show that "1 in 100" outweighs "1 per cent"; they show that presentation shifts the weight under some conditions.
Yuval Rottenstreich and Christopher Hsee, at the University of Chicago, argued in 2001 that the curve's shape depends on what is at stake. Outcomes of equal money value differ in how much emotion they evoke, they proposed, and the weighting curve is more S-shaped for emotionally rich outcomes: people are more sensitive to moving away from impossibility and from certainty, and less sensitive to changes in the middle. In their test a prize low in emotion was preferred to a prize high in emotion when each was certain, and the preference reversed when each carried a low probability.
Kahneman reports, without naming it, a later test by a Princeton team in which a rich, vivid picture of an outcome reduced the weight given to its probability whether or not the outcome was emotional. On that account vividness, not emotion, is the operative variable, and a 1 per cent chance of something easy to picture would carry more weight than a 1 per cent chance of something abstract.
His endnotes appear to point to a 2010 paper by A. Peter McGraw, Eldar Shafir and Alexander Todorov in Management Science, whose abstract reports that evaluations of non-monetary outcomes respond less to probability than evaluations of money. We read only that abstract, and it does not state the vividness conclusion, so that conclusion rests on Kahneman's account alone. An article on [denominator neglect and the bigger bowl](https://thewhyweproject.com/articles/denominator-neglect-and-the-bigger-bowl) lays out why a count can outweigh a ratio.
Sampling the odds instead of reading them
Ralph Hertwig and colleagues tested the reverse case in 2004. One hundred students at the Technion in Haifa took six problems whose rare events had probabilities of 0.2, 0.1 or 0.025. Half saw the outcomes and probabilities described on a screen. The other half saw two unlabelled buttons, each tied to a hidden payoff distribution, and pressed them as often as they liked before choosing, with a median of 15 draws per problem. The chosen option was played for real money.
The two groups chose differently, by an average of 36 percentage points. In every one of the six problems the direction fitted the claim that rare events are underweighted when they are experienced and overweighted when they are described, and all but one of the differences were significant. The authors tested two explanations for the underweighting, small samples and extra weight on recent draws, and concluded that risky choice needs two theories, one for each mode. Kahneman puts the mechanism plainly: many participants never experience the rare event.
A reanalysis, and Kahneman's revision
A 2006 reanalysis of the same data disputes that conclusion. It found that the experience group's choices fitted prospect theory in 69 per cent of cases once the theory was applied to the probabilities the participants had actually experienced, against 27 per cent when it was applied to the stated probabilities.
A replication with 46 students at Ben-Gurion University found that judged and experienced probabilities correlated at a median of .97. There, 63 per cent of choices conformed to prospect theory when it was applied to each participant's judged probabilities, and 60 per cent departed from it when it was applied to the objective ones. The reanalysis reads the apparent underweighting as mostly sampling error, because small samples tend to miss low-probability outcomes, and calls the claim of two theories premature.
A 2013 study found that the gap persists even when participants sample the whole population of outcomes. In a repeated-choice task that controlled for sample bias, underweighting persisted in choices while overweighting appeared in judged probabilities, and the authors concluded that accounts assuming overweighting of small probabilities do not describe decisions from experience even without sampling error.
Kahneman's own account moved. In the chapter on rare events he writes that the original theory said highly unlikely events are either ignored or overweighted but did not specify the conditions under which one or the other will occur, and he gives attention as the missing condition. A 1 per cent chance weighs about 5.5 beside a vivid outcome, and may carry little or nothing when a sample misses it or the event never comes to mind.
Common questions
Why does a 1 per cent chance feel bigger?
Because the mind gives a possible outcome more weight than its odds warrant. In Tversky and Kahneman's 1992 estimates a 1 per cent chance carries a decision weight of about 5.5. The weight rises when the outcome is vivid, as Rottenstreich and Hsee proposed in 2001, and Kahneman reports that vividness is the operative variable.
Do people always overweight a 1 per cent chance?
No. In Hertwig and colleagues' 2004 experiment, people who learned rare events by sampling behaved as if they underweighted them, and their choices differed from those of people who read the odds by an average of 36 percentage points. Kahneman also writes that below 1 per cent no unique decision weight can be assigned, and a 2006 reanalysis traced the gap mainly to sampling error rather than to a second theory, a reading that a 2013 study disputes.
Is the 5.5 weight true of everyone?
It is a median estimate for gains from 25 graduate students choosing among monetary prospects, from the 1992 paper, and Tversky and Kahneman report medians, not a constant for every person. Whether it holds for a hazard, a lottery ticket or a side effect is a separate question that the 1992 experiment did not test.
Sources
Kahneman, D. (2011). Thinking, Fast and Slow. New York: Farrar, Straus and Giroux. Chapters 29 and 30, pp. 311 to 333, as recorded in the vault's digests of the book.
Kahneman, D., & Tversky, A. (1979). 'Prospect Theory: An Analysis of Decision under Risk.' Econometrica, 47(2), 263-291. doi:10.2307/1914185
Tversky, A., & Kahneman, D. (1992). 'Advances in Prospect Theory: Cumulative Representation of Uncertainty.' Journal of Risk and Uncertainty, 5(4), 297-323. doi:10.1007/BF00122574
Rottenstreich, Y., & Hsee, C. K. (2001). 'Money, Kisses, and Electric Shocks: On the Affective Psychology of Risk.' Psychological Science, 12(3), 185-190. doi:10.1111/1467-9280.00334. Read at the level of the abstract.
Hertwig, R., Barron, G., Weber, E. U., & Erev, I. (2004). 'Decisions from Experience and the Effect of Rare Events in Risky Choice.' Psychological Science, 15(8), 534-539. doi:10.1111/j.0956-7976.2004.00715.x
Fox, C. R., & Hadar, L. (2006). 'Decisions from Experience = Sampling Error + Prospect Theory: Reconsidering Hertwig, Barron, Weber & Erev (2004).' Judgment and Decision Making, 1(2), 159-161. doi:10.1017/S1930297500002370.
Barron, G., & Ursino, G. (2013). 'Underweighting Rare Events in Experience Based Decisions: Beyond Sample Error.' Journal of Economic Psychology, 39, 278-286. doi:10.1016/j.joep.2013.09.002. Read at the level of the abstract.
Slovic, P., Monahan, J., & MacGregor, D. G. (2000). 'Violence Risk Assessment and Risk Communication: The Effects of Using Actual Cases, Providing Instruction, and Employing Probability Versus Frequency Formats.' Law and Human Behavior, 24(3), 271-296. Read at the level of the abstract.
Monahan, J., Heilbrun, K., Silver, E., Nabors, E., Bone, J., & Slovic, P. (2002). 'Communicating Violence Risk: Frequency Formats, Vivid Outcomes, and Forensic Settings.' International Journal of Forensic Mental Health, 1(2), 121-126. doi:10.1080/14999013.2002.10471167. Read at the level of the abstract, which also summarises the 2000 study by Slovic, Monahan and MacGregor.
McGraw, A. P., Shafir, E., & Todorov, A. (2010). 'Valuing Money and Things: Why a $20 Item Can Be Worth More and Less Than $20.' Management Science, 56(5), 816-830. doi:10.1287/mnsc.1100.1147.