# Children’s concept of probability as inferred from their binary choices—revisited

## Abstract

Children had to choose one of two urns—each comprising beads of winning and losing colours—from which to draw a winning bead. Three experiments, aimed at diagnosing rules of choice and designed without confounding possible rules with each other, were conducted. The level of arithmetic difficulty of the trials was controlled so as not to distort the effects of the constituent variables of proportion. Children aged 4 to 11 first chose by more winning elements and proceeded with age to choices by greater proportion of winning elements. There were some indications of intermediate choices by fewer losing elements and by greater difference between the two colours. Distinguishing correct choices from favourable draws, namely acknowledging the role of chance in producing the outcome and insisting on the right choice, grew with age. Children switched rather early from considering one dimension to two; they combined the quantities of winning and losing elements either additively by difference or, with age, mostly multiplicatively by proportion. Guided playful activities for young children, based on this research, are suggested for fostering acquisition of the basic constituents of the probability concept: uncertainty of outcome, quantification by proportion, and the reverse relation between the chances of complementary events.

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## Notes

1. 1.

The apparent certainty of success following a correct choice in Noelting’s (1980a, b) task is due to the law of large numbers. The sampled beverage for drinking comprises multitude of molecules. Were the participants to draw a single molecule, the situation would reduce to that of drawing a bead from an urn.

2. 2.

Quoted in Teaching Statistics (1985), 7, 92

3. 3.

The drop in the functions for greater w at age 8 might be accidental or a case of the “U-shaped behavioral growth” in proportional reasoning, as observed by Stavy et al. (1982).

4. 4.

Though Bayesian analysis implies some increase (decrease) of the confidence in the correctness of a choice following even one instance of positive (negative) feedback, this does not entail certainty.

5. 5.

The story goes that Bertrand Russell commented on this apparent oxymoron: “How dare we speak of the laws of chance? Is not chance the antithesis of all law?”

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## Acknowledgements

The study was partly supported by the Sturman Center for Human Development, The Hebrew University of Jerusalem. We are grateful to Raphael Falk for his invaluable help in all the stages of the research and to graduate students of the Hebrew University for their assistance in collecting data for Experiment 3. Yael Oren took care of the figures.

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Correspondence to Ruma Falk.

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Falk, R., Yudilevich-Assouline, P. & Elstein, A. Children’s concept of probability as inferred from their binary choices—revisited. Educ Stud Math 81, 207–233 (2012). https://doi.org/10.1007/s10649-012-9402-1

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### Keywords

• Comparing probabilities
• Probability and proportion
• Diagnostic design
• Relevant-involvement methodology
• Recognizing chance
• One- versus two-dimensional reasoning