Table 1 From left to right each column corresponds with: dataset on which statistics are computed, local P-value for the null hypothesis RNG A is uniform, local P-value for the null hypothesis RNG B is uniform, local P-value for the null hypothesis RNG A&B is uniform, Fisher’s test, Pearson’s test and p_threshold and joint P-values p_joint.

dataset		n ₀₀	n ₀₁	n ₁₀	n ₁₁	n	RNG A	RNG B	RNG A&B	Pearson	Fisher	p _threshold	p _joint
Run 1	All recorded data	4938847	4942101	4939328	4942337	19762613	0.872	0.159	0.568	0.956		0.016	0.144
	Bell trials	53	79	62	51	245	0.250	0.371	0.054		0.029	0.021	0.121
Run 2	All recorded data	4529615	4530943	4528295	4526440	18115293	0.171	0.901	0.486	0.455		0.016	0.144
	Bell trials	69	69	78	84	300	0.184	0.773	0.545		0.817	0.018	0.131
Bell trials of both runs combined		122	148	140	135	545	0.864	0.392	0.452		0.211	0.020	0.138

The joint P-value for a set of hypotheses is the probability that for at least one of the hypotheses we observe a P-value less than α where here α = 0.05. This captures the fact that the more hypotheses we test, the more likely it becomes that one of them will fall below the significance threshold. The value p_threshold the largest threshold for individual tests for which the joint P-value for that row is less than 0.05. The local P-values in the row should this be compared to this number. This captures the fact that when testing multiple hypothesis, the local P-values of the individual ones actually need to be much smaller for the overall test to be significant. The local P-values in columns RNG A, RNG B, Fisher and Pearson are exact calculations. The columns RNG A&B, p_threshold and p_joint are approximations obtained via 10⁵, 10⁴ and 10⁴ trials of a Monte-Carlo simulation, respectively.

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