TAdults can make quick decisions about probability: if there’s a 90% chance of rain, they should bring an umbrella. However, is this ability innate to human beings? At what point in childhood might this ability emerge, independently of formal education?
Given a small sample containing a 1:4 ratio, can 8-month-old infants infer the population based on the probabilistic sample? Experimenters (Xu & Garcia, 2008) brought a covered box to the table containing ping pong balls inside (either 70 red and 5 white, or 70 white and 5 red – the child did not get to see the distribution of the balls). The experimenter would reach inside the box and remove a ball, saying “Look!” and placing the ball in front of the child. They did this 5 times until there was either a distribution of 4 red and 1 white, or 4 white and 1 red. Then, they removed the cover of the box to reveal the distribution of the population and measured the child’s looking time. Authors hypothesized that if the child saw 4 red and 1 white, they would look longer at the unexpected outcome of 70 white and 5 red. This should be reversed for the 1 red and 4 white condition. Results showed that infants indeed looked longer at the unexpected outcome.
Can infants reason about probability in large set sizes? Denison and Xu (2010) tested 32 12-14-month-old infants in a probabilistic task involving choosing the location of the preferred lollipop. Infants indicated through crawling which color lollipop they liked, pink or black. Then, in the test trial, the infant saw two clear boxes, each with a distribution of lollipop colors. One box had more pink lollipops, and one box had more black lollipops. The experimenter let the child observe both boxes, and then reached into one box and took out a lollipop, using their hand to occlude the color of the lollipop. They put this lollipop in an opaque cup. They then reached into the other box and took out a lollipop, also occluding this with their hand, and placed it in another cup. Results showed that in 78% of trials, infants crawled or walked to the cup that had the higher probability of having their preferred color lollipop. This study demonstrates that infants ages 12-14 months can make probabilistic decisions about relatively large quantities.
However, Teglas et al.’s (2015) results are different, suggesting that intuitions of probabilities are derived from representations of a limited, not large, number of outcomes. In experiment 1 of 2, authors tested 20 12-month-old infants in a lottery box task. In the test trial, in the machine were 3 objects of 1 category and 1 object of a different category bouncing around. Then, an occluder covered the machine, and then one object exited the container. After that, the occluder disappeared to show the container and the remaining objects. Infants saw four of these trials, two with the majority object exiting, and two with the single object exiting. Findings reveal that infants looked longer when the single object exited the machine.
Are infants paying attention to the number of objects (altogether 4) or the ratio of objects (3:1) in order to predict which object would exit the machine? Authors tested these two claims in experiment 2. Most of the stimuli and the movies remained the same, except that there were 16 objects with a ratio of 3:1 (either 12 blue, 4 yellow or 12 yellow, 4 blue). In this experiment, infants looked at either outcomes in non-significantly different durations. Authors suggested that the scene containing 16 objects was simply too complex for infants to analyze and they were overwhelmed by the situation. Infants either could not follow the number of objects or they were not able to deduce the ratio of objects – either scenario could be possible and does not help explain whether infants succeeded in the first experiment due to counting the objects or observing the ratio. In either case, given a small size set, infants can make probabilistic decisions, but they are unable to do so when a large set size is presented.
If infants can make choices based on probability (Xu & Garcia, 2008; Denison & Xu, 2010; Teglas et al., 2015), can young children ages 3-5 years do the same? Girotto et al. tested 93 children ages 3-5. On a table there are two puppets, two piles of chips of two colors (with differing sizes and differing distributions of red and black chips), two opaque boxes, two opaque mugs, some black stickers, and some red stickers. The experimenter asked the child to choose the puppet they like more. The child and their puppet were placed in the red team, which won if they received more red stickers. The other puppet was in the black team, which won if it received more black stickers. The child was then told that they receive a red sticker every time they find a red chip. Then, the experimenter showed the child the two piles of chips, putting each pile into an opaque box. On each trial, the experimenter would pick a chip from a box and place it in the mug, doing the same to the other box. The child was then asked to choose the cup that would have a red chip so as to win a red sticker. There were 5 trials in total, each with different piles of chips, increasing with difficulty in the proportion of red or black chips in each box.
In trial 1, when one pile had all red chips and one pile had all black chips, all children performed well. In all other trials except one, however, the youngest age group (3-year-olds) performed poorly, guessing at or below chance. In trial 4, a highly demanding trial (proportions were close but amounts were the same), all children performed above chance. However, in the most demanding task (the two piles had largely differing sizes but the smaller pile was more favorable), both 3 and 4-year-olds performed below chance. It would appear that 3 and even 4-year-olds don’t use probability in choice tasks consistently. It is possible that 3-year-olds follow the absolute probability heuristic, always choosing the larger pile (if there is one), whereas 4-year-olds follow that heuristic when probabilities become closer to each other. 5-year-olds are able to overcome this heuristic and make decisions based on probability.
Placi et al. (2020) attempt to address the issue of the development of probabilistic decision-making in infancy to preschool age. 68 children participated and were split into two age groups: infants (10-12 months, 34 children) and preschoolers (36-54 months, 34 children). Transparent white and blue Kinder eggs were attached to wooden sticks. In all blue eggs were finger puppets representing different animals, and all white eggs were empty. There were two transparent boxes covered by blankets. Two opaque cups were used for occluding the color egg picked. In a preference trial, the experimenter introduced the child to the eggs and showed them what was inside. Then, they put each egg into a cup and asked the child to choose the one they liked. Only children who picked the cup with the blue egg were retained for the test trial.
In the test trial, the experimenter filled one box with a favorable ratio of blue eggs to white eggs (9:3) and another with an unfavorable ratio (16:48). The experimenter then shook the box to mix the eggs, then chose an egg at random with box boxes and placed them into the cup in front of the box. They indicated that the child picked the one they want. If the child relied on absolute quantities, they would pick the egg from the unfavorable box. If the child relied on probability, they would pick the egg from the favorable box. Results demonstrated that 50% of infants selected the egg from the favorable box and 53% of preschoolers selected the egg from the favorable box; both are no different from chance. These results suggest that neither infants nor preschoolers used probability to make inferences about what might be in the cups.
Both Xu and Garcia (2008) and Teglas et al. (2015) found that infants could make probabilistic decisions on small set sizes. However, Denison and Xu (2010) found that infants could also make probabilistic decisions on large set sizes (20), while Teglas et al. did not find the same results on a large set size (16). This could have happened due to object tracking. In Xu and Garcia (2008) and Denison and Xu (2010), the stimuli remained stationary (in boxes or placed in front of the child). In Teglas et al.’s study (2015), objects bounced around in a machine. It has been shown that children cannot keep track of more than 4 objects. In the studies where stimuli were stationary (Xu & Garcia, 2008; Denison & Xu, 2010), children could see the distribution of objects. In Teglas et al.’s (2015) study, objects moved around and then were occluded. Perhaps children were able to keep track of 4 objects in the first experiment but failed when there were 16 objects. It would appear in this case that object tracking has hindered children’s ability to make probability judgments.
Given that infants appeared to be able to make probabilistic decisions (Xu & Garcia, 2008; Denison & Xu, 2010), it is odd that infants failed in Placi et al. (2020). A possible explanation for infants’ failure was that they did not have a strong preference for the blue eggs, considering that 45% of the original sample chose the white egg in the preference trial and that it is possible that the children who remained also chose between the eggs at chance, just happening to pick the blue one. Another explanation for the failure of infants is that in the lollipop task (Denison & Xu, 2010), parents saw the experiment and could have influenced their child, whereas parents wore opaque glasses in this experiment. As for the preschoolers, perhaps in both Placi et al. (2020) and Girotto et al. (2016), the 3-4-year-olds faced higher cognitive load than the infant studies, which did not have overly complicated ratios that required comparisons. 3-4 years is an interesting age for development as children have shown to be capable if task demands are not overly complicated (for example, the gap in 3-4-year-olds in theory of mind tasks). It is possible that task demands in these two experiments (playing a game: Girotto et al., 2016; winning puppets: Placi et al., 2020) were too much for the preschoolers, but that an easier task might be devised where 3-4-year-olds can succeed.
This nutshell offers only a minimal peak into the complex field of probabilistic reasoning in children. The aforementioned studies hold several diverging perspectives and the question remains: when do children start being able to reason about probability, and when can they do it well?
Denison, S., & Xu, F. (2010). Twelve- to 14-month-old infants can predict single-event probability with large set sizes. Developmental Science, 13(5), 798–803. https://doi.org/10.1111/j.1467-7687.2009.00943.x
Girotto, V., Fontanari, L., Gonzalez, M., Vallortigara, G., & Blaye, A. (2016). Young children do not succeed in choice tasks that imply evaluating chances. Cognition, 152, 32–39. https://doi.org/10.1016/j.cognition.2016.03.010
Placi, S., Fischer, J., & Rakoczy, H. (2020). Do infants and preschoolers quantify probabilities based on proportions? R. Soc. Open Sci. 7. 191751. http://dx.doi.org/10.1098/rsos.191751
Téglás, E., Ibanez-Lillo, A., Costa, A., & Bonatti, L. L. (2015). Numerical representations and intuitions of probabilities at 12 months. Developmental Science, 18(2), 183–193. https://doi.org/10.1111/desc.12196
Xu, F., & Garcia, V. (2008). Intuitive Statistics by 8-Month-Old Infants. Proceedings of the National Academy of Sciences of the United States of America, 105(13), 5012–5015.