By L.D. Meshalkin

The Russian model of a set of difficulties in likelihood concept encompasses a bankruptcy dedicated to facts. That bankruptcy has been passed over during this translation simply because, within the opinion of the editor, its content material deviates a little bit from that that is instructed via the identify: difficulties in professional bability concept. the unique Russian model includes a few mistakes; an try out was once made to right all error stumbled on, yet maybe a number of stiII stay. An index has been extra for the benefit of the reader who will be looking for a definition, a classical challenge, or no matter what. The index lists pages in addition to difficulties the place the listed phrases look. The booklet has been translated and edited with the wish of leaving as a lot "Russian style" within the textual content and difficulties as attainable. Any pecu liarities current are probably as a result this goal. August, 1972 Bryan A. Haworth viii Foreword to the Russian version This choice of difficulties in chance thought is basically meant for collage scholars in physics and arithmetic departments. Its target is to assist the scholar of chance concept to grasp the speculation extra seasoned foundly and to acquaint him with the applying of chance conception the way to the answer of useful difficulties. This assortment is geared essentially to the 3rd variation of the GNEDENKO textbook direction in proba bility concept, Fizmatgiz, Moscow (1961), chance conception, Chelsea (1965).

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**Extra info for Collection of problems in probability theory**

**Example text**

Hint. Introduce the conditional probabilities u and v of the event A under the conditions that the results of the first trial are respectively ace and non-ace. Using the formula for total probability, set up the equations relating u and v. 148. * Consider a sequence of independent trials each with three possible outcomes A, B, C of corresponding probabilities p, q and r (p + q + r = I). 33 Application of the basic formulas Find the probability that a) a series of ()( consecutive A-outcomes occurs earlier than the first series of f3 consecutive B-outcomes; b) a series of ()( consecutive A-outcomes occurs before one gets either f3 consecutive B-outcomes or y consecutive C-outcomes.

In this case N analyses are needed. 2. The blood of k persons is mixed and the mixture obtained is analyzed. If the result of the analysis is negative, then this single analysis is sufficient for k persons. But if it is positive, then the blood of each one must be subsequently investigated separately, and in toto for k persons, k+ 1 analysis are needed. It is assumed that the probability of a positive result is the same for all persons and that the results of the analyses are independent in the probabilistic sense.

Entropy is measured in units corresponding to the logarithm base a. In this problem book, we take a=2. Suppose given two random variables ~ and tf, and let Pij=Pg=X i ; tf=yJ; then we define H(~, tf), the entropy of the random variables ( and tf, as - L Pu 10gaPu . j and the conditional entropy I H (( tf = yj) = 1) See also page 71. 38 L P (i Ij) loga P {i Ij}. i Calculation of mathematical expectations The quantity is called the mean conditional entropy of ~ relative to rJ, and the quantity I~ (~) = H (~) - H (~ 111) is called the quantity of information, contained in rJ relative to ~.