Probability and distributions

Experiments and Events

Definitions:

• An experiment is a process that leads to a single outcome that cannot be predicted with certainty. The set of all possible results of an experiment is called the support of the experiment (usually called, Omega, W).
• A simple event is an outcome of an experiment that cannot be decomposed into single outcomes. An event is a collection of one or more simple events of interest. Events are generally symbolized by upper case letters, A, B, …., etc.
• Two events are mutually exclusive if they cannot occur at the same time.

Two events are independent events if the occurrence of one does not affect the occurrence of the other.

Examples of experiments:

• Toss a coin once and observe the up face;
• There are two possible results: Head (H), Tail (T);
• W = {H, T}; Simple events: {H} and {T}
• Gender of a baby before the normal conception (single baby);
• There are two possible results: male (M), female (F);
• W = {M, F}; Simple events: {M} and {F}
• Toss a dice once and observe the up face;
• There are 6 possible results: : 1, 2, …., 6;
• W = {1, 2, 3,   , 6}; Simple events: {1}, {2},    {6};
• The result of tossing a dice is greater than 2: E = {3, 4, 5, 6}.

Probability

Definitions:

• The probability of an event E, P(E): percentage of chances that event E be realized.
  • Suppose that an experiment is repeated a large number of times. Each repetition is called a trial. Suppose that for each trial, there is a certain event of interest, E. Then,

P(E) is in fact a relative frequency, as seen in descriptive statistics.