Denominator Neglect and the Bigger Bowl
Ten in a hundred and one in ten are the same probability, yet people reach for the larger count. Bowls of beans, a paid marble task, a disease rated riskier at half the death rate and a study of forensic clinicians show why, and what happens when the pull reverses.
Answer to the question
Why does 10 in 100 feel likelier than 1 in 10?
They are the same probability, ten per cent. The felt difference comes from attending to the number of cases and underweighting the total they sit within, a pattern called denominator neglect. In Epstein's bowl experiments, many people preferred nine winning beans in a hundred to one in ten, though the small bowl had the better odds.
Key takeaways
- Ten in a hundred and one in ten are the same probability, ten per cent, so any felt difference comes from the count of cases and not from the odds.
- In the 1994 bowl study by Denes-Raj and Epstein, as later researchers report it, 61 per cent chose nine red beans in a hundred over one in ten, and 40 per cent still chose seven in a hundred.
- Across 300 rounds in Dale and colleagues' 2007 study, participants picked the small urn 59.0 per cent of the time when it was right and the large urn 75.5 per cent when it was right, so errors leaned towards the bigger count.
- Yamagishi (1997) found a disease killing 1,286 in 10,000 judged riskier than one killing 24.14 in 100, which is 12.86 per cent against 24.14.
- Price and Matthews (2009) showed that stating only the number affected reversed the group-diffusion effect into a ratio bias, so which number is made salient decides the result.
In the ratio-bias experiments run in Seymour Epstein's laboratory, people are shown two bowls of beans and told they win if they draw a red one blind. The small bowl holds ten beans and one is red. The large bowl holds a hundred and nine are red. Many reach for the large bowl, and that choice is denominator neglect in its plainest form. The small bowl is the better bet, ten per cent against nine. Denominator neglect is the habit of attending to the number of cases, nine red beans, and giving too little weight to the base they sit on, a hundred.
It is also called the ratio bias, and Daniel Kahneman uses the label in Thinking, Fast and Slow. It also answers a question people type into search engines: why does ten in a hundred feel likelier than one in ten? The two are the same probability. Whatever difference a reader feels comes from the count of cases, ten against one, and not from the odds.
The objection to take seriously comes first. The people who pick the large bowl may simply be careless, or unable to do the sum, and a result built on carelessness would tell us little. The evidence on that is more interesting than either side expects.
Bowls of beans and the odds people know but ignore
Nine winners in a hundred against one in ten was the contest in the 1994 version, and 61 per cent of participants, as later researchers report it, chose the larger bowl. Against seven in a hundred, 40 per cent still did. The paper, by Denes-Raj and Epstein, followed a first demonstration by Kirkpatrick and Epstein in 1992, and its title states the finding: conflict between intuitive and rational processing, in which people behave against their better judgment.
Some participants said they knew the small bowl gave better odds and drew from the large one anyway. Epstein's account of denominator neglect, cognitive-experiential self-theory, is that an experiential system runs alongside a rule-based one and does not defer to it, which is why the conflict can be felt as well as inferred. At ten in a hundred against one in ten, the sum cannot separate the bowls at all, so anyone who feels the larger bowl is likelier is responding to the numerator.
Three hundred rounds of choices, some of them paid
Money is the obvious test of carelessness, and in 2007 Dale, Rudski, Schwarz and Smith ran it, asking whether denominator neglect holds up when mistakes cost money. Participants made 300 choices between a ten-marble urn and a hundred-marble urn, under six conditions that ran from no payment to five or ten cents for every red marble drawn. The bias was there, and it could be told apart from general random error.
When the small urn was the right choice, participants picked it 59.0 per cent of the time. When the large urn was the right choice, they picked it 75.5 per cent of the time. Random slips would fall evenly on both sides. These fell on the side where the count of winners was larger.
Money helped in the within-subject comparison, though the between-subject tests were not significant. The bias also shrank over the rounds while overall accuracy stayed put. Carelessness is a poor explanation for a pull that leans one way and fades only with experience.
Random slips would fall evenly on both sides. These fell on the side where the count of winners was larger.
A disease rated more dangerous at half the death rate
The laboratory bowls would matter little if denominator neglect stopped at beans. Kimihiko Yamagishi showed in 1997 that it does not. A disease described as killing 1,286 people out of every 10,000 was judged riskier than one that kills 24.14 out of every 100. The first is 12.86 per cent and the second 24.14, so the disease rated as more dangerous had about half the death rate.
The most consequential version was run on professionals. Paul Slovic and colleagues gave forensic psychologists and psychiatrists case summaries of patients hospitalised with mental disorders and asked them to judge the risk of violence after discharge. At the same level of likelihood, a patient looked more dangerous when the risk was stated as ten out of a hundred than as ten per cent, and two out of ten sounded far worse than twenty per cent. The authors link the gap to the more frightening images that frequencies call up.
A disease described as killing 1,286 people out of every 10,000 was judged riskier than one that kills 24.14 out of every 100.
Kahneman gives the figures in his book without naming the researchers: clinicians refused discharge in 41 per cent of cases when the risk was framed as a frequency and 21 per cent when it was framed as a probability. His description matches the study by Slovic and colleagues described above, though the published abstract does not state these percentages, so they stand here as his report.
Anyone who writes a risk figure for others is choosing a denominator, whether they notice or not, and the same honest statistic can be dressed to frighten or to reassure. Format is one route by which a small chance gets inflated. [Why a one per cent chance feels bigger](https://thewhyweproject.com/articles/why-a-1-per-cent-chance-feels-bigger) follows another, through the weights people give to probabilities, and [a loss and a gain of the same size](https://thewhyweproject.com/articles/loss-aversion-and-the-bet-you-turn-down) do not weigh the same either.
Salience decides which of the two numbers wins
The group-diffusion effect looks like the opposite of all this. It is the tendency to rate your own risk as lower when more people share the same threat. Paul Price and Teri Matthews argued in 2009, in Judgment and Decision Making, that the two are one process. Both come from judging ratios, and the result depends on which term is more salient: the denominator, the total exposed, gives group diffusion, and the numerator, the number affected, gives ratio bias.
Their experiments moved the effect on demand. Stating the number expected to be affected removed the group-diffusion effect. Stating only that number reversed it into a ratio bias.
That sits awkwardly with an account in which denominator neglect is a fixed trait of people who are permanently drawn to the vivid count. Epstein's two-system reading describes the felt conflict well, and it fits the people who knew the odds and chose against them. The salience account does more work, because it predicts when the pull will weaken or flip, and we side with it as the economical explanation, with the two-system story left standing as a description of what the conflict feels like.
In the fourth of Price and Matthews's tests, handing people the number affected without the total was enough to turn an effect that makes risk feel smaller into one that makes it feel larger. Which number comes first also shapes the judgement that follows, as [the first price you see](https://thewhyweproject.com/articles/why-the-first-price-you-see-decides-what-you-pay) shows. The salience account rests on one paper of four experiments, so it is the better explanation on offer and not a settled one.
Common questions
Why does 10 in 100 feel likelier than 1 in 10?
They are the same probability, ten per cent. The felt difference comes from attending to the larger count of cases and underweighting the base. In a study of forensic clinicians by Paul Slovic and colleagues, ten out of a hundred looked more dangerous than ten per cent, and two out of ten far worse than twenty per cent.
What is denominator neglect?
Denominator neglect is giving too much weight to the number of cases in a ratio and too little to the total they sit within. It is also called the ratio bias. In Epstein's bowl experiments, many people preferred nine winning beans in a hundred to one in ten, though the small bowl had the better odds.
Is denominator neglect the same as base-rate neglect?
No. In base-rate neglect the base is missing from the story, and a vivid description pushes it aside. In denominator neglect the base is stated in the ratio and still gets too little weight.
Sources
Kahneman, D. (2011). Thinking, Fast and Slow. New York: Farrar, Straus and Giroux
Kirkpatrick, L. A., & Epstein, S. (1992). 'Cognitive-experiential self-theory and subjective probability: Further evidence for two conceptual systems.' Journal of Personality and Social Psychology, 63(4), 534-544
Denes-Raj, V., & Epstein, S. (1994). 'Conflict between intuitive and rational processing: When people behave against their better judgment.' Journal of Personality and Social Psychology, 66(5), 819-829
Dale, D., Rudski, J., Schwarz, A., & Smith, E. (2007). 'Innumeracy and incentives: A ratio bias experiment.' Judgment and Decision Making, 2(4), 243-250
Yamagishi, K. (1997). 'When a 12.86% mortality is more dangerous than 24.14%: Implications for risk communication.' Applied Cognitive Psychology, 11, 495-506
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. doi:10.1023/A:1005595519944
Price, P. C., & Matthews, T. V. (2009). 'From group diffusion to ratio bias: Effects of denominator and numerator salience on intuitive risk and likelihood judgments.' Judgment and Decision Making, 4(6), 436-446