Ruma Falk

Some teasers concerning conditional probabilities

A family of notorious teasers in probability is discussed. All ask for the probability that the objects of a certain pair both have some property when information exists that at least one of them does. These problems should be solved using conditional probabilities, but cause difficulties in characterizing the conditioning event appropriately. In particular, they highlight the importance of determining the way information is being obtained. A probability space for modeling verbal problems should allow for the representation of the given outcome and the statistical experiment which yielded it. The paper gives some psychological reasons for the tricky nature of these problems, and some practical tips for handling them.

Children's construction of fair chances: Adjusting probabilities.

A probability-adjustment task was presented to 6-14-year-old children. In 2 experiments, children had to generate equal probabilities by completing the missing beads in a target urn with 1 type of beads presented beside a full urn with both winning and losing beads. The task was embedded in a competitive game. This relevant-involvement method secured optimal understanding and motivation. The analysis was based on number of matches with, and sum of distances from, the correct response and the predictions of other strategies. The results indicate that only at around the age of 13 did most children proportionally integrate the 2 dimensions (i.e., the numbers of winning and losing beads). The youngest sometimes relied on 1 dimension, and 9- and 10-year-olds partly combined the 2 types of quantities additively. The cognition involved in probability adjustment was analytic rather than global or intuitive. The ability to generate equal probabilities is discussed in terms of the many faces of probability.

A potential for learning probability in young children

Sixty-one children, from 4 to 11 years old, were presented with two sets, each containing blue and yellow elements. Each time, one colour was pointed out as the payoff colour (POC). The child had to choose the set from which he or she would draw at random a POC element in order to be rewarded. The sets were of varying sizes with different proportions of the two colours. The problem was to select the higher of the two probabilities. Three kinds of materials were used: Pairs of urns with blue and yellow beads, pairs of roulettes divided into blue and yellow sectors, and pairs of spinning tops, likewise divided into two colours.Roughly around the age of six, children started to select the greater of the two probabilities systematically. The dominant error was selecting the set with the greater number of POC elements. Verbal concepts of probability and chance were explored and some ‘egocentric’ thought processes were described. The study indicates that probability concepts could be introduced into school teaching even in the first grades. The deterministic orientation in the instruction for young ages should be attenuated, permitting concepts of uncertainty right from the beginning.

The allure of equality: Uniformity in probabilistic and statistical judgment ☆

Uniformity, that is, equiprobability of all available options is central as a theoretical presupposition and as a computational tool in probability theory. It is justified only when applied to an appropriate sample space. In five studies, we posed diversified problems that called for unequal probabilities or weights to be assigned to the given units. The predominant response was choice of equal probabilities and weights. Many participants failed the task of partitioning the possibilities into elements that justify uniformity. The uniformity fallacy proved compelling and robust across varied content areas, tasks, and cases in which the correct weights should either have been directly or inversely proportional to their respective values. Debiasing measures included presenting individualized and visual data and asking for extreme comparisons. The preference of uniformity obtains across several contexts. It seems to serve as an anchor also in mathematical and social judgments. Peoples pervasive partiality for uniformity is explained as a quest for fairness and symmetry, and possibly in terms of expediency.

The exchange paradox: Probabilistic and cognitive analysis of a psychological conundrum

The term “exchange paradox” refers to a situation in which it appears to be advantageous for each of two holders of an envelope containing some amount of money to always exchange his or her envelope for that of the other individual, which they know contains either half or twice their own amount. We review several versions of the problem and show that resolving the paradox depends on the specifics of the situation, which must be disambiguated, and on the players beliefs. The latter psychological variables are part and parcel of the resolution. Assuming reasonable subjective distributions, exchanging cannot always be advantageous for both players. We suggest several deep-rooted psychological reasons for the considerable difficulty people demonstrably have in dealing with this problem. Implicit widespread and compelling assumptions—that affect judgement in diverse contexts—obstruct the solution. Analysing this paradox underscores the close connection between psychology and probability theory.

The Infinite Challenge: Levels of Conceiving the Endlessness of Numbers

To conceive the infinity of integers, one has to realize: (a) the unending possibility of increasing/decreasing numbers (potential infinity), (b) that the cardinality of the set of numbers is greater than that of any finite set (actual infinity), and (c) that the leap from a finite to an infinite set is itself infinite (immeasurable gap). Three experiments probed these understandings via competitive games and choice tasks accompanied by in-depth interviews. Participants were children 6 to 15 years old and adults. The results suggest that roughly from about age 8 on, children grasp potential and actual infinity. However, for several additional years their conception of actual infinity is incomplete because the immeasurable gap between a finite and an infinite set is not entirely internalized. Even many adolescents and adults fail to appreciate this gap. Distinguishing between number concepts and their names facilitates conceiving aspects of infinity. Educational implications of these findings are discussed.

Tell Me the Method, I'll Give You the Mean

Different procedures underlying the determination of the mean of a set of values impart different weights to these values. Thus, different methods result in different weighted means. In particular, when the weights are directly proportional, or in fact equal, to the (weighted) values, the self-weighted mean is obtained, and when they are inversely proportional to the values, the harmonic mean results. Generally, the former is greater and the latter is smaller than the arithmetic (uniformly weighted) mean. We illustrate cases of sampling procedures and of geometric requirements that give rise to the self-weighted and harmonic means. We call attention to some psychological difficulties and to the didactic value of dealing with cases of differential weighting.

Average Speed Bumps: Four Perspectives on Averaging Speeds

highway, where each car travels at a constant speed. Assume that the distribution of the speeds is the same throughout the length of the highway. You adjust your speed so that during a given time unit you overtake the same number of cars as the number of cars that overtake you. Does this mean that your speed is the median of the speeds of the cars on the highway? Surprisingly, the answer is no! (Clevenson, Schilling, Watkins, & Watkins, 2001). Imagine further a radar device at the side of the highway, measuring and recording the speeds of all the cars that pass this point within a fixed time interval. Again, contrary to lay expectations, the arithmetic mean of these recordings would generally not reproduce the arithmetic mean of the speeds of all the cars on the highway (Stein & Dattero, 1985). The above examples illustrate that identifying the correct average may have its difficulties (the correct answers for both cases will be detailed later). Average speed, in general, is not all that self-evident a concept. The apparently simple question “what is the average speed of the cars that drive on the highway?” is equivocal. As students of introductory statistics know, the term average may be interpreted in various ways and hence may assume several different forms. One needs to know in what sense the average is supposed to represent a set of observations. Four Faces of the Average Speed

Lewis Carroll’s Obtuse Problem

Carroll’s apparently impeccable solution to one of his probability problems is shown to answer another problem that is based on reasonable assumptions. His original assumptions, however, are self contradictory, hence entailing paradoxical results.

The Surprisingness of Coincidences1

Abstract Coincidence-type stories, a popular social pastime, were studied in terms of factors affecting their surprisingness. Subjects were presented with stories varying in tense (past vs. future) and scope of detail (union vs. intersection). The normal form of coincidence stories, past-intersection, was found to be judged less surprising than stories of identical scope told in the future tense and similar in surprisingness to past-tense stories of greater scope. Subjects judged their own coincidences as more surprising than coincidences constructed by the experimenters and more surprising than when evaluated by a separate group of subjects. Implications of these results are discussed in terms of what we may remember or learn from experiences related by others.