Theory of Probability The probability of an event is expressed on a scale from 0.0 to 1.0

0.0 means it will never happen

1.0 means that it is certain to happen The probability of an event is the number of times that a specific event occurs relative to the sum of all possible events that can occur. Example The probability of rolling a 3 on a die is 1 out of 6, or 0.1667 Example The probability of two coins flipped once coming up heads is 1 out of 4, or 0.25 Theoretical probability Sometimes we know from the theory of the matter what the probability of an event is, e.g. rolling dice or flipping coins.

Relative frequency probability In other cases we can only estimate the probability by observing how frequently a particular event occurs, e.g. jury acquittals, people quitting a job

Terminology Statistical experiment An empirical record of any phenomenon whose relative frequency of occurrence is uncertain. Example Data on sentencing outcomes: probability of acquittal, fine, deferred adjudication, probation, or prison Sample space All possible outcomes, Example Possible sentencing outcomes: acquittal, fine, deferred adjudication, probation, or prison An event A specific outcome or collection of outcomes; e.g. an acquittal A complement All possible events other than the one in question, e.g. if the event in question is acquittal, then the complements are fine, deferred adjudication, probation, and prison

Terminology (cont.)

Probability of an event The proportion of times an event occurs divided by the frequency of all other events that can occur, e.g. 246 acquittals out of 14,573 cases acquitted, fined, deferred, probated, or sent to prison (246/14,573 = p = 0.0169) Relative frequency How often an event occurs relative to all other events that occurred in the experiment Mutually exclusive events Two or more events which can not happen together, .e.g. acquitted and sent to prison, the probability = 0.0 Conditional probability The probability of event A happening, given that event B has already occurred, e.g. probability of going to prison (A) given that the offender was put on probation (B).

This is symbolized P(A B)

Terminology (cont.)

Independent events Two events A & B are considered independent if the conditional

probability P(A B) = P(A), e.g. probability of acquittal (A) given that it is raining outside (B)

Addition Rule of Probability

Q What is the probability of A or B happening?

If the two events are not mutually exclusive …

P(A or B) = P(A) + P(B) – P(A and B)

Example What is the probability of drawing either a Jack or a Heart from a deck of cards? P(J or H) = P(J) + P(H) – P(J and H) P(J or H) = P(4/52) + P(13/52) – P(1/52) P(J or H) = (0.0769 + 0.2453) – (0.01923) P(J or H) = 0.3077

Addition rule (con’d)

If the two events are mutually exclusive …

P(A or B) = P(A) + P(B)

Example What is the probability of drawing a Jack or a King?

P(J or H) = P(J) + P(H) P(J or H) = P(4/52) + P(4/52) P(J or H) = (0.0769 + 0.0769) = 0.1538

Multiplication Rule of Probability

Q What is the probability of A and B happening

together?

The general rule

P(B) P(AB) If the events are independent of each other, this simplifies to …

P(A) P(B)

Example What is the probability of drawing a Jack of Hearts? Since J and H are not mutually exclusive, therefore independent …

P(J and H) = P(J) P(H) = (4/52) (13/52) = 0.01923

Notice that this is the same as (1/52) = 0.01923