Reasoning Under Uncertainty: Maximum Likelihood Heuristic in a Problem With a Random Transfer

Figures & data

Fig. 1 Example for a trial in the comparison of probabilities task.

Fig. 2 Example of a trial in the additional condition of uncertainty task.

Table 1 Description of the trials classified according to $P_b(A)$ and the formal correct judgment.

	Type of trial
	Type-A		Type-E		Type-B
	Urn B	Urn A	Urn B	Urn A	Urn B	Urn A
$P_b(A)$	black/white	black/white	black/white	black/white	black/white	black/white
$P_b(A)<0.5$	1/2	2/3	4/6	2/3	2/2	2/3
	2/6	1/2	2/4	1/2	2/3	1/2
	2/3	3/4	1/2	2/4	2/2	2/4
	3/5	2/3	2/6	1/3	4/6	2/4
	1/3	3/5	1/3	2/6	3/4	2/3
$P_b(A)=0.5$	2/4	2/2	3/3	2/2	4/2	2/2
	3/4	3/3	2/2	3/3	4/3	3/3
	3/5	3/3	1/1	3/3	5/3	3/3
	2/5	2/2	4/4	3/3	5/2	2/2
	2/3	2/2	4/4	2/2	3/2	2/2
$P_b(A)>0.5$	2/2	3/2	6/4	3/2	2/1	3/2
	3/2	2/1	4/2	2/1	6/2	2/1
	6/4	4/2	2/1	4/2	3/2	4/3
	2/2	4/2	6/2	3/1	5/3	3/2
	4/3	3/2	3/1	6/2	3/1	5/3

NOTE: The numbers in the table represent the ratio of black to white balls in each trial.

Table 2 Comparison of the accuracy of responses between the two tasks (N = 66).

		Accuracy in percent (%) of responses: Mean (SD)
$P_b(A)$	Trial	CP task	ACU task	t(65)
$P_b(A)<0.5$	Type-A	69.4 (27.0)	33.9 (33.8)	6.6***
	Type-E	64.2 (37.4)	13.9 (25.5)	8.9***
	Type-B	71.8 (24.0)	59.7 (34.4)	2.9**
$P_b(A)=0.5$	Type-A	82.4 (24.2)	33 (30.1)	9.5***
	Type-E	84.8 (27.6)	57.9 (36.5)	4.9***
	Type-B	75.2 (30.8)	22.7 (26.0)	5.6***
$P_b(A)>0.5$	Type-A	69.4 (25.6)	68.5 (34.6)	0.2
	Type-E	58.8 (35.9)	12.7 (22.0)	9.0***
	Type-B	61.21 (30.8)	22.7 (26.0)	7.8***

NOTE: Paired t-tests were employed to compare the accuracy of responses between the two tasks.

${}^{**}p<$ 0.01.

${}^{***}p<$ 0.001.

Table 3 Associations of $P_b(A)$ and the correct judgment with participants’ responses in each task.

		CP task		ACU task
Participants’ responses	Comparison/reference	OR Group I	OR Group II	OR
Urn A/Will increase	$P_b(A)>0.5/P_b(A)=0.5$	1.02	2.28***	10.87***
	$P_b(A)>0.5/P_b(A)<0.5$	1.04	1.47	1.86**
	Type-A/Type-E	202.92***	38.04***	1.89***
	Type-A/Type-B	5.92***	2.8***	0.8
Equal/Will not change	$P_b(A)=0.5/P_b(A)>0.5$	1.02	2.36***	11.00***
	$P_b(A)=0.5/P_b(A)<0.5$	1.30	2.19***	6.35***
	Type-E/Type-A	231.27***	42.026***	1.89***
	Type-E/Type-B	113.19***	36.62***	2.4***
Urn B/Will decrease	$P_b(A)<0.5/P_b(A)>0.5$	1.30	1.32	6.29***
	$P_b(A)<0.5/P_b(A)=0.5$	1.30	2.15***	14.55***
	Type-B/Type-A	19.8***	8.00***	1.96*
	Type-B/Type-E	98.364***	33.20***	1.82**

${}^{*}p<$ 0.05.

${}^{**}p<$ 0.01.

${}^{***}p<$ 0.001.

Table 4 Distribution of participants’ responses in the additional condition of uncertainty task according to two specific compositions of Urn A, with several different compositions of Urn B (for the same Urn-A composition) and the formal correct judgment.

			Formal correct judgment: Probability of drawing a black ball after the transfer	Participants’ response (%): Probability of drawing a black ball from Urn B after the transfer
Urn A	Urn B			Will	Will	Will
black/white	black/white	$P_b(A)$		increase	not change	decrease
2/3	2/2	$P_b(A)<0.5$	Will decrease	24.8	10.7	54.5
2/3	3/4		Will decrease	25.8	6.1	68.1
2/3	1/2		Will increase	34.9	12.1	53
2/3	3/5		Will increase	31.8	6.1	62.1
2/3	4/6		Will not change	22.7	16.7	60.6
3/2	2/2	$P_b(A)>0.5$	Will increase	68.2	13.6	18.2
3/2	4/3		Will increase	68.2	12.1	19.7
3/2	2/1		Will decrease	71.2	12.1	16.7
3/2	5/3		Will decrease	62.1	7.6	30.3
3/2	6/4		Will not change	59.1	21.2	19.7

NOTE: Most frequent responses in each comparison are marked with bold.

Table 5 Comparison of the reaction times in milliseconds between the two tasks.

	Formal correct judgment: CP task/ACU task	Task reaction time (ms)
$P_b(A)$		CP: Mean (SD)	ACU: Mean (SD)	t(65)
$P_b(A)<0.5$	Urn A/Will increase	7168 (4360)	4366 (5313)	3.4***
	Equal/Will not change	5409 (3206)	3295 (3609)	3.9***
	Urn B/Will decrease	6252 (3796)	4191 (4235)	3.3***
$P_b(A)=0.5$	Urn A/Will increase	5034 (2884)	4818 (5576)	0.3
	Equal/Will not change	3183 (2609)	5905 (4500)	4.7***
	Urn B/Will decrease	5103 (3147)	4708 (5634)	0.6
$P_b(A)>0.5$	Urn A/Will increase	7704 (4722)	4889 (5721)	3.3***
	Equal/Will not change	6847 (4489)	3270 (4927)	5.0***
	Urn B/Will decrease	7087 (5195)	3371 (4116)	5.1***

NOTE: Paired t-tests were employed to compare reaction time between the two tasks. Each participant’s reaction time was calculated as the median of the reaction times (for both correct and incorrect responses) in each type of trial.

${}^{***}p<$ 0.001.