# Chapter4.ppt - Chapters 4 Probability Theory u2013 Study...

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1 Chapters 4 Probability Theory – Study of Randomness
2 Idea of probability A phenomenon is random if individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions. Probability of an outcome of a random phenomenon is the proportion of time it would occur in a very long series of repetitions.
3 Basic definitions Sample space -- the set of all possible outcomes of a random phenomenon. Event a subset of the sample space Probability model Description of sample space A way to assign probabilities to events
4 Assigning probabilities to sample spaces Assign a probability to each individual outcome. Each probability is a number between 0 and 1 and the probabilities must sum to 1. The probability of any event is the sum of the probabilities of the outcomes making up the event.
5 Basic Probability Rules and Definitions The probability of any event is between 0 and 1. If S is the sample space, then P(S) =1. The complement of an event A is the event, A c , that A doesn’t occur. P(A c ) = 1- P(A)
6 Basic Probability Rules and Definitions Two events, A and B, are disjoint if they have no elements in common and, therefore, can’t occur simultaneously. We then have P(A or B) = P(A or B occurs) =P(A) + P(B)
7 Addition rule for several disjoint events If A, B, and C are disjoint events then P(at least one of A, B, C ) = P(A) +P(B) + P(C ). This extends to any number of disjoint events.
8 Independent Events Two events, A and B, are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, then P(A and B) = P(A)P(B) This is called the multiplication rule for independent events.