Conditional Probability

Conditional probability is based on the probability of an event occuring after a previous event has already occured. The probability of the second event depends on the first. The | is used to indicate conditional probability. If two events, A and B, occur one after the other, but the probability of B depends on A then it is written as:

P(B|A)

which is read as "the probability of B given A".

When both A and B occur it is indicated by: P(A ∩ B)

The probability of A occuring is: P(A)

Since the probability of B occuring given A depends on A occuring first we write:

P(A ∩ B) / P(A) represents the probability of the outcome containing A and B coming from A.

The formula above can also be written as:

P(B|A)P(A) = P(A ∩ B) by cross multiplying.

Total Probability

If a sample space S is divided into n partitions and a subset B occurs in the sample space, the total probability of B can be found by adding all the occurences of B in each partition.

P(B) = P(B|S₁)P(S₁) + P(B|S₂)P(S₂) + P(B|S₃)P(S₃) + P(B|S₄)P(S₄)

For n partitions this would be:

P(B) = ∑ⁿ_{i = 1}P(B|S_i)(S_i)

The probabilities of B in each partition are added together to get the probability of B.

Bayes' Theorem

Bayes' theorem allows probabilities to be calculated from total probabilities. If S is the sample space and S₁...S_n are the partitions and B is a subset of S then:

The probability of the partitioned space can be found based on a subset of that space.