Refresher - Probability

DEF: The sample space S for a probability model is the set of all possible outcomes.

DEF: An event A is a subset of the sample space S. (Possible outcome(s))

DEF: A probability is a numerical value assigned to a given event ${\displaystyle A}$. The probability of an event is written ${\displaystyle P\left(A\right)}$, and describes the long-run relative frequency of the event.

If there are k possible outcomes for a phenomenon and each is equally likely, then each individual outcome has probability 1/k.

The probability of any event E is given by:

Axioms of Probability:

• Rule 1: Any probability ${\displaystyle P\left(A\right)}$ is a number between 0 and 1  .
• Rule 2: The probability of the sample space S is equal to ${\displaystyle 1P\left(S\right)=1}$.
• Rule 3: If two events A and B are disjoint, then the probability of either event is the sum of the probabilities of the two events: .
• Rule 4: The probability that any event A does not occur is ${\displaystyle P\left(A^c\right)=1-P\left(A\right)}$.
• Rule 5: If two events A and B are independent, then the probability of both events is the product of the probabilities for each event: .
• Rule 6: If ${\displaystyle \varnothing }$ is the empty set, then ${\displaystyle P(\varnothing )=0}$