Probability— the relative frequency or likelihood that a specific event will occur. If the event is A, then the probability that A will occur is denoted P(A). Example: Flip a coin. What is the probability of heads? This is denoted P(heads). Properties of Probability 1. The probability of an event E always lies in the range of 0 to 1; i.e., 0 ≤ P( E ) ≤ 1. Impossible event—an event that absolutely cannot occur; probability is zero. Example: Suppose you roll a normal die. What is the probability that you will get a seven? P(7) = 0. Sure event—an event that is certain to occur; probability is one. Example: Suppose you roll a normal die. What is the probability that you will get a number less than 7? P(a number less than 7) = 1. 2. The sum of the probabilities of all simple mutually exclusive events (or final outcomes) that can occur in a population or sample events in an expirement is always 1. Example: Suppose you flip two coins. What are the outcomes? HH, HT, TH, TT. This rule says that the probabilities of each of these outcomes should sum to one. That is, P(HH) + P(HT) + P(TH) + P(TT) = 1 Marginal and Conditional Probabilities Suppose the faculty at a local school were polled as to their agreement/disagreement with the following statement: Coaches should be paid more than regular classroom teachers. The following two-way (contingency) table contains the results. AGREE DISAGREE MALE 20 10 FEMALE 15 35 From such a table, we can compute two types of probability—marginal and conditional. First, you should add a row and column to the table for totals.
AGREE DISAGREE TOTAL MALE 20 10 30 FEMALE 15 35 50 TOTAL 35 45 80
Marginal Probability—the probability of a single event without consideration of any other event; also called simple probability. Example: P(male) = (# of males)/(total # of teachers) = 30/80 Example: P(agree) = (# of teachers who agree)/(total # of teachers) = 35/80 Conditional Probability—the probability that an eve nt will occur given that another event has already occurred. Example: Suppose that one teacher is selected at random. It is known that the teacher is a male. What is the probability that the teacher agrees? P( agrees male) = (# of males who agree) /(total # of males ) = 20 / 30 .
P( agrees male) is read “the probability the teacher agrees given that the teacher is a male.” Example: P( male agrees ) = (# of males who agree ) /( total # who agree) = 20 / 35 . ________________________________________________________________________ Mutually Exclusive Events & Independent/Dependent Events Mutually Exclusive Events—events that cannot occur together. Example: A card is chosen at random from a standard deck of 52 playing cards. Consider the event “the card is a 5” and the event “the card is a king.” These two events are mutually exclusive because there is no way for both events to occur at the same time. Non-example: A card is chosen at random from a standard deck of 52 playing cards. Consider the event “the card is a heart” and the event “the card is a king.” These two events are not mutually exclusive because it is possible for these two events to occur at the same time, namely when the King of Hearts is selected. Independent Events—two events are independent if the occurrence of one does not affect the probability of the occurrence of the other. If A and B are independent events (that is, the knowledge or occurrence of one does not affect the probability of occurrence of the other), then by definition P( A B ) = P( A) and P( B A) = P( B). Dependent Events—two events are dependent if the occurrence of one affects the occurrence of the other.
Example: Consider the following two-way table.
YES MALE 15 FEMALE 4 TOTAL 19
NO TOTAL 45 60 36 40 81 100
Are the events “female” and “yes” independent? Does P( female yes ) = P( female) ?
P( female yes ) = 4 / 19 = 0.211 P(female) = 40/100 = 0.4 These events are dependent. In terms of the problem, this means that the...