Probability Topics: Terminology 1.4 2008/05/08 15:11:46 GMT-5 2008/07/31 10:11:08.317 GMT-5 Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Connexions cnx@cnx.org A AND B A OR B chance conditional equally likely event experiment frequency key terms long term outcome Probability relative sample space Statistics Terminology This module defines several key terms related to Probability. Probability measures the uncertainty that is associated with the outcomes of a particular experiment or activity. An experiment is a planned operation carried out under controlled conditions. If the result is not predetermined, then the experiment is said to be a chance experiment. Flipping one fair coin is an example of an experiment.The result of an experiment is called an outcome. A sample space is a set of all possible outcomes. Three ways to represent a sample space are to list the possible outcomes, to create a tree diagram, or to create a Venn diagram. The uppercase letter S is used to denote the sample space. For example, if you flip one fair coin, S = {H, T} where H = heads and T = tails are the outcomes.An event is any combination of outcomes. Upper case letters like A and B represent events. For example, if the experiment is to flip one fair coin, event A might be getting at most one head. The probability of an event A is written P(A).The probability of any outcome is the long-term relative frequency of that outcome. For example, if you flip one fair coin from 20 to 2,000 times, the relative frequency of heads approaches 0.5 (the probability of heads). Probabilities are between 0 and 1, inclusive (includes 0 and 1 and all numbers between these values). P(A) = 0 means the event A can never happen. P(A) = 1 means the event A always happens.To calculate the probability of an event A, count the outcomes for event A and divide by the total outcomes in the sample space. For example, if you toss a fair dime and a fair nickel, the sample space is {HH, TH, HT, TT} where T = tails and H = heads. The sample space has four outcomes. A = getting one head. There are two outcomes {HT, TH}. P(A) = 2 4 .Equally likely means that each outcome of an experiment occurs with equal probability. For example, if you toss a fair, six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face.An outcome is in the event A OR B if the outcome is in A or is in B or is in both A and B. For example, let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. A OR B = {1, 2, 3, 4, 5, 6, 7, 8}. Notice that 4 and 5 are NOT listed twice.An outcome is in the event A AND B if the outcome is in both A and B at the same time. For example, let A and B be {1, 2, 3, 4, 5} and {4, 5, 6, 7, 8}, respectively. Then A AND B = {4, 5}.The complement of event A is denoted A' (read "A prime"). A' consists of all outcomes that are NOT in A. Notice that P(A) + P(A') = 1. For example, let S = {1, 2, 3, 4, 5, 6} and let A = {1, 2, 3, 4}. Then, A' = {5, 6}. P(A) = 4 6 , P(A') = 2 6 , and P(A) + P(A') = 4 6 + 2 6 = 1 The conditional probability of A given B is written P(A|B). The probability of A is calculated knowing that B has already occurred. A conditional reduces the sample space. We calculate the probability of A from the reduced sample space B. The formula to calculate P(A|B) isP(A|B)= P(A AND B) P(B) where P(B) is greater than 0.For example, suppose we toss one fair, six-sided die. The sample space S = {1, 2, 3, 4, 5, 6}. Let A = face is 2 or 3 and B = face is even (2, 4, 6). To calculate P(A|B), we count the number of outcomes 2 or 3 in the sample space B = {2, 4, 6}. Then we divide that by the number of outcomes in B (and not S). We get the same result by using the formula. Remember that S has 6 outcomes.P(A|B)= P(A and B) P(B) = (the number of outcomes that are 2 or 3 and even in S) / 6 (the number of outcomes that are even in S) / 6 = 1/6 3/6 = 1 3 Conditional Probability The likelihood that an event will occur given that another event has already occurred. Equally Likely Each outcome of an experiment has the same probability. Experiment A planned activity carried out under controlled conditions. Event A subset in the set of all outcomes of an experiment. The set of all outcomes of an experiment is called a sample space and denoted, as a rule, by S. An event is any arbitrary subset in S: it can contain one outcome, two outcomes, and even no outcomes (empty subset) or all of them (sample space). Standard notations for events are capital letters such as A, B, C, etc. Outcome (observation) A particular result of an experiment. Probability A number between 0 and 1, inclusive, that gives the likelihood that a specific event will occur. More exact, the foundation of statistics are given by the following 3 axioms (by A. N. Kolmogorov, 1930’s): Let S denote the sample space, A and B are any two events in S . Then: (1). 0≤P(A)≤1; size 12{0 <= P \( A \) <= 1;} {} (2). If A and B are any two mutually exclusive events, then P ( A or B ) = P ( A ) + P ( B ) ; (3). P ( S ) = 1 . Sample Space The set of all possible outcomes of an experiment.