Well if such a sequence is rare, then heads is due

by Andrea Elizabeth

From the Cambridge Dictionary of Philosophy

gambler’s fallacy, also called Monte Carlo fallacy, the fallacy of supposing, of a sequence of independent events, that the probabilities of later outcomes must increase or decrease to “compensate” for earlier outcomes. For example, since (by Bernoulli’s theorem) in a long run of tosses of a fair coin it is very probable that the coin will come up heads roughly half the time, one might think that a coin that has not come up heads recently must be “due” to come up heads – must have a probability greater than one-half of doing so. But this is a misunderstanding of the law of large numbers, which requires no such compensating tendencies of the coin. The probability of heads remains one-half for each toss despite the preponderance, so far; of tails. In the sufficiently long run what “compensates” for the presence of improbably long subsequences in which, say, tails strongly predominate, is simply that such subsequences occur rarely and therefore have only a slight effect on the statistical character of the whole. See Bernoulli’s theorem

Bernoulli’s theorem is too complicated. The entry before that on St. Bernard of Clairvaux is more interesting:

(1090 – 1153), French Cistercian monk, mystic, and religious leader. He is most noted for his doctrine of Christian humility and his depiction of the mytical experience, which exerted considerable influence on later Christian mystics. Educated in France, he entered the monastery at Citeaux in 1112, and three years later founded a daughter monastery at Clairvaux.

According to Bernard, honest self-knowledge should reveal the extnt to which we fail to be what we should be in the eyes of God. That self-knowledge should lead us to curb our pride and so become more humble. Humility is necessary for contemplation of God, the highest form of which is union with God. Consistent with orthodox Christian doctrine, Bernard maintains that mystical union does not entail identity. One does not become God; rather, one’s will and God’s will come into complete conformity. See mysticism.

Not sure it’s Orthodox according to St. Athanasius.

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