By David Stirzaker

This totally revised and up to date new version of the good demonstrated textbook gives a transparent advent to the idea of chance. themes coated contain conditional likelihood, independence, discrete and non-stop random variables, producing features and restrict theorems, and an creation to Markov chains. The textual content is obtainable to undergraduate scholars and offers a variety of examples and routines to aid increase the real abilities invaluable for challenge fixing. First version Hb (1994): 0-521-42028-8 First version Pb (1994): 0-521-42183-7

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Because of the symmetry of a perfect die, we assume that these 36 outcomes are equally likely. 1) the required probability is 6 1 |A| = = . | | 36 6 (b) There are six outcomes of the form (i, 7 − i) whose sum is 7, so the probability that 6 = 16 . the sum is 7 is 36 There are two outcomes whose sum is 11—namely, (5, 6) and (6, 5)—so the prob2 1 = 18 . ability that the sum is 11 is 36 1 = 29 . 3), the required probability is 16 + 18 (c) It is routine to list the outcomes that do have a common factor greater than unity.

First, consider an experiment with equally likely outcomes, for which we deﬁned probability as the proportion P(A) = |A| . | | If A and B are disjoint then, trivially, |A ∪ B| = |A| + |B|. Hence, in this case, P(A ∪ B) = |A ∪ B| |A| |B| = + = P(A) + P(B). | | | | | | Second, consider the interpretation of probability as reﬂecting relative frequency in the long run. Suppose an experiment is repeated N times. At each repetition, events A and B may or may not occur. If they are disjoint, they cannot both occur at the same repetition.

36 36 Doing it this way gives a slightly quicker enumeration than the direct approach. 1− Worked Examples and Exercises (1) (2) (3) (4) (5) 41 Exercise What is the probability that the sum of the numbers is 2, 3, or 12? Exercise What is the probability that: (a) The sum is odd? (b) The difference is odd? (c) The product is odd? Exercise What is the probability that one number divides the other? Exercise What is the probability that the ﬁrst die shows a smaller number than the second? Exercise What is the probability that different numbers are shown and the smaller of the two numbers is r, 1 ≤ r ≤ 6?