The Nature of Probability

• An experiment is a controlled operation that yields a set of results.
• The possible results of an experiment are called outcomes.
• An event is a collection of outcomes from an experiment.
• Empirical probability is the relative frequency of occurrences of an event by actual observation of an experiment.
• Theoretical probability is determined through study of the possible outcomes that can occur for a given experiment.
• P(E) = #times E occurs/ # times the experiment was performed.
• The Law of Large numbers states that probability statements apply in practice to a large number of trials, not to a single trial. It is the relative frequency of the long run that is accurately predictable, not individual events or precise totals.

Theoretical Probability

• If each outcome of an experiment has the same chance of occurring as any other outcome, then the outcomes are equally likely.
• P(E) = n(E) /N(U) if all outcomes are equally likely.
• P(E) = 0, then E can not occur
• P(E) = 1, then E must occur
• 0 ≤ P(E) ≤ 1 for all E
• The sum of the probability of all outcomes of an event must be 1.
• P(A) + P(not A) = 1
• P(A) = 1- P(not A)

Odds

• Odds and probability are directly related, but odds are not probability and probability is not odds.
• Odds are either odds in favor of E or odds against E.
• Odds in favor of E are P(E)/P(not E)
• Odds against E are P(note E)/P(E)