Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
FOUNDATION MATHEMATICS 042 Review of Elementary Probability Definitions and Properties (Random) experiment. A method by which observations are made, e.g. one can have the experiment of rolling a die once, to observe the number of dots on its uppermost face. Outcome. A possible observation of an experiment, e.g. 1 is an outcome of the above experiment. Sample space. The set of all outcomes of an experiment, e.g. {1, 2, 3, 4, 5, 6} is the sample space of our example experiment. Event. A set of outcomes of a random experiment, e.g. the event that one rolls an even number is the set {2, 4, 6}. A simple event is a set containing a single outcome of a random experiment. With the following properties of sets and probability functions it is recommended that you draw for yourself corresponding Venn diagrams. Probability function. P is a probability function on a sample space S if • 0 ≤ P (A) ≤ 1 for every event A of S; • P (S) = 1; and • P (A ∪ B) = P (A) + P (B) for every pair of disjoint events A, B of S. A and B are disjoint if their intersection is empty, i.e. A ∩ B = ∅. Note that it follows from above that • P (A ∩ B) = 0 if A ∩ B = ∅. Suppose Xi represent the outcomes of a random experiment. Then, to show that P is a probability function on its sample space S, it is enough to show that (i) P (Xi ) ≥ 0 for each outcome Xi ∈ S, and P (ii) i P (Xi ) = 1. Complementary event. The complement of an event A in a sample space S, is the set of all outcomes in S that are not in A. We denote this complementary event of A, by A (or sometimes by A0 ). De Morgan’s Laws. • A∪B =A∩B • A∩B =A∪B 1 Correspondence ∪ ↔ or, ∩ ↔ and. • A ∪ B is the event that A or B occurs. • A ∩ B is the event that A and B occur. Mutually exclusive. Two events A, B are mutually exclusive if A, B are disjoint as sets. Independent. Two events A, B are independent if the probability of one event occurring is unaffected by whether or not the other event has occurred. Conditional probability. It may happen that the sample space S is effectively reduced to a subset B of S. In this case we may say the probability of an event A given B and write P (A | B). B here is the “condition”. Further properties of probability functions. Let P be a probability function on the sample space S. Then we have the following properties where A, B are events in S. • Complementary events A and A satisfy: P (A) = 1 − P (A) • The conditional probability P (A | B) is given by: P (A | B) = P (A ∩ B) P (B) • If A and B are independent events then P (A | B) = P (A) and P (B | A) = P (B) so that we have P (A ∩ B) = P (A | B).P (B), = P (A).P (B) by rearranging the conditional probability rule • For any events A, B in S, P (A ∪ B) = P (A) + P (B) − P (A ∩ B) Hence, if A, B are mutually exclusive, so that P (A ∩ B) = 0, then P (A ∪ B) = P (A) + P (B) 2