Table 2.

Probability that the response is correct if it is driven by an intention (2^nd column), by automaticity based on probability matching (3^rd column), or reactive control (4^th column) depending on the trial type. In this example there are 70% AX trials, 10% AY trials, 10% BX trials, and 10% BY trials. The first term of the accuracy of proactive control (e.g., $\frac{7}{8}$) is the probability that the person predicted the more likely probe when they set their intention. The second term is the loss in accuracy due to cognitive load. load is the intensity of the cognitive load given by the number of different letters that could serve as the A-cue (i.e., 1, 2, or 3).

Cue, probe	Accuracy of proactive control P(CI = 1|  cue, probe, load)	Accuracy of automaticity P(CM = 1| cue, probe)	Accuracy of reactive control P(CR = 1| cue, probe, load)
AX	$\frac{7}{8}-\lambda ·\left(\text{load}-1\right)$	$P\left(\mathit{AX}|X\right)=\frac{7}{8}$	1 − λ · (load − 1)
AY	$\frac{1}{8}-\lambda ·\left(\text{load}-1\right)$	P(AX′| Y) = 1	1 − λ · (load − 1)
BX	1 − λ · (load − 1)	$1-P\left(\mathit{AX}|X\right)=\frac{1}{8}$	1 − λ · (load − 1)
BY	1 − λ · (load − 1)	P(AX′| Y) = 1	1 − λ · (load − 1)