By L. S. Pontryagin, V. G. Boltyanskii

ISBN-10: 2881240771

ISBN-13: 9782881240775

The fourth and ultimate quantity during this finished set offers the utmost precept as a panoramic option to nonclassical, variational difficulties. This one mathematical technique should be utilized in quite a few events, together with linear equations with variable coefficients, optimum methods with hold up, and the bounce . As with the 3 previous volumes, the entire fabric contained with the forty two sections of this quantity is made simply obtainable in terms of a variety of examples, either concrete and summary in nature.

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I l ^ j ) . Now and/? 2)= 1 - P r ( / ? , | / ? 2 ) = 1- ~ = ^ - Neither P r ( / ? i | / ? 2 ) nor P r ( / ? 2 l ^ i ) can be found without knowing the composition of the balls at the start. Lacking this information, other, perhaps unwarranted, assumptions must be made. Our next example brings this out. Example 5 A bag contains two balls, each of which may be either red or white. A ball is drawn at random and found to be red. What is the probability that the other ball is red? To start with, the bag could contain 0, 1 or 2 red balls.

Since the total number of patients is 50, 3 0 - M C n ^ ) + 3 6 < 5 0 , or N(Cr)A)>\6. Problem 3. For the men, 80 = Ν(Μ') + N(Af) = N(M' η C') + N(M' η Ο + N{M) = 26 + Ν{Μ' Π Q + 44. Hence Ν{Μ' Π Q = 80 — 70 = 10, and the required number of women is 25 — 10 = 15. The total number of women is here redundant information. Problem 4. x{E η G ) - Pr(£ Π F Π G ) . For the last line, consult Problem 1, part (1). Problem 5. 1. = 720. = 96. Problem 6. 3" + 1 = 82. Problem 7. rt, + 1 of colour /• (including none).

3 Conditional Probability and Independence 62 other assumptions one might make, a plausible one being that each ball has equal probability of being white or red. This means that WW, RW, WR, RR are to have equal probability. T(R) = 0 X - 1 + 4 1 - X 2 - + 1 2 X 1 = 4 1 2 1 Pr(//o\R) = 0 , Pr(//, \R)= - , 1 P r ( / / j \ R ) = \ - ? τ { Η , \R)= - . When so appHed, ( 3 . 4 ) is known as Bayes' formula Example 6 We now return to Example 4 , and take it one stage further. That is, after first draw has been completed, a second ball is drawn at random, if white replaced together with another white, if red it is replaced together with more reds.

### The mathematical theory of optimal processes by L. S. Pontryagin, V. G. Boltyanskii

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