Conditional Probability
Basic Probability Notes
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
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