Probability Models and Conditioning

The sample space (omega)collectively exhaustive for the experimentmutually exclusiveright scope ‘granularity’

The probability lawassigns a probability to every event in the

sample spacesatisfies the axioms of probability

Probability Models

Nonnegativity: P(A)>=0 for all AAdditivity: If A and B are disjoint sets, then

the probability of their union is the sum of the probabilities of the set.

Normalization: P(omega)=1

All other equations relating to probability laws are ultimately derived from the above axioms.

Kolmogorov’s Axioms of Probability

A shift into a new, related sample space, with a new probability lawThe new law still satisfies the axioms of

probabilityThe definition on page 20 could be written in

words, “Whatever even A occurs must occur within the event B, so B is now the entire sample space.”

relative probabilities are maintainedAlternatively, the conditional probability

could be viewed as remaining on the same sample space omega, but normalized by B.

Conditioning

Partition the sample space into A1…Anthen P(B) is the probability of B found in each

set Ai summed over all iThis amounts to P(B) being a weighted

average of P(B|Ai) summed over all i, with P(Ai) as the weighting factor

Total Probability Theorem

The probability of a series of events is the product of the probability of each event in the series, conditioned on the previous event

Serial probability

A way to infer the probability of an event Ai as a cause, out of i possible causes for event B, which has been observed.We want to know the probability of Ai|BB has been observed, so the probability of the

B also falling within the event Ai should be the ratio of the the probability of B occurring and being in event Ai (i.e. P(A)*P(B|Ai)) to the probability of B occurring at all.

Verifiable by the definition of conditional probability (see pg. 20 and pg 32).

Bayes Rule (the simple, discrete version)