Probability

The collection of outcomes possible from an experiment can be represented as a set.

For example, in the case of a die the sample space can be written:

Ω = {1,2,3,4,5,6}

You may be betting on an Event, that is a particular subset of outcomes.

For example, you may be betting that the result of a die will be an even number, ie that the result will be a member of the subset:

E = {2,4,6}

The likelihood of the result of a throw being the empty set Ø={} is 0, the likelihood of Ω is certain.

An elementary event corresponds to a single sample point. The elementary event {5} is a subset of Ω whereas 5 is a member of Ω.

The cardinality (number of members) of a set is written as |A|. It is obvious that for dice where the likelihood of numbers is equal:

P(A) = |A| / |Ω|

Two events are independent if P(B|A)= P(B)

A random variable can be assigned to different values. The notation { X=r } refers to a random variable with name X and value r.

If there is a limited number of possible outcomes, then the the outcome is a discrete random variable.

Events are said to be exhaustive if the union of the sets includes every possible sample point.

Events are said to be exclusive the the associated sets are disjoint.

P({X=r}) means the probability of the random variable X being r.

P(E) means the probability of the event E.

Say you are rolling the dice twice, the first result being labelled r and the second s. You can write the sum as:

P(B|A) = Probability of B given A =

from medicine.mcgill.ca