Probability

Gist

A gambler's favorite topic

Axioms

A probability space consists of a sample space S and a probability function P which takes an event A ⊆ S as input and returns P(A), a real number between 0 and 1, as output. The function P must satisfy the following axioms:

Properties:

P(Ac) = 1 − P(A).
If A ⊆ B, then P(A) ≤ P(B).
P (A ∪ B) = P (A) + P (B) − P (A ∩ B).

P(A ∩ B) = P(A) * P(B|A)

The joint probability that the event A and B are to occur is that the

Other Laws

Chain Law of Probability

The joint probability of a sequence of events can be computed by multipliying the conditional probability of an even given the preceding event. If we have events $A_{1}, A_{2}, . . ., A_{n}$ . The probability of $A_{n}$ happening is:

P (A_{1}, A_{2}, . . ., A_{n}) = P (A_{1}) P (A_{2} | A_{1}) P (A_{3} | A_{1}, A_{2}) . . . P (A_{n} | A_{1}, . . . A_{n - 1})

Used for large-language models. If I want to predict the probability of the sentence I love cheese then it would be:

P (I, L o v e, C h e e s e) = P (I) P (L o v e | I) P (C h e e s e | I, L o v e)

References

Introduction ot PRobability