Probability.md (1566B)
1 # Probability 2 3 Stats CH1 4 5 **Definition:** The probability is the likelihood of something happening as a percentage between 0 and 1 or 0% and 100%. 6 7 Let X be a set and F a set of subsets of X. A probability on (X,F) is a function u : F -> [0,1]. This means for each set in F we have a probability between 0 and 1 for each set. See [SetFunction](SetFunction.md) for more about the u (mu greek character) function. 8 9 The probability function must be a set function, but that is not sufficient. We also need for u(0) where 0 is the empty set to be equal to 0. We also need u(X) = 1 (totaling 100%), and if A and B are disjoint sets then u(A union B) = u(A) + u(B). This final part means the probability of the union of two different sets is equal to the sum of the probabilities of both sets individually. 10 11 When we have a domain that is finite we then state we have a [DiscreteProbability](DiscreteProbability.md) whereas when we have an interval then the function is said to be a [[ContinuousProbability.md]]. 12 13 In practical terms, for u(X) X is the set off outcomes that are possible and the function returns the probability of said outcome. 14 15 **Sometimes we use u({1}) but often we use P(1) to describe the probability of 1. Notice that u({1}) takes in a set whereas P(1) does not require such statements.** 16 17 Some syntax for ya, when stating P(H|Theta = 1/3) this means the probability of H given that theta = 1/3. This is often used when we don't know the probability of theta, but need to describe the situation. In this instance, theta is considered a parameter.