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142. Assuming that, in the multiplication of bacteria, the probability that a bacterium divides into two new bacteria in the interval of time r equals unmatnr) and does not depend on the number of bacteria 32 Setting up equations present, not- on the number of preceding divisions, nd the probability that, there are i bacteria present at time t if there was one bacterium at time (I. 143. Continuation. Let us asSume in addition that, independently of its own preceding historyr and of the total number of bacteria present, a bacterium alive at time i will perish in the time interval (t, r+ A!) with probability ogli+o{,,t}. Set up the di'erential equations which the probabilities 11(1), that there are r bacteria at time i, must satisfy. 144. n mechaniSms are switched in an electric power transmission line. The probability that a mechanism requiring energy at time i terminates its requirement at the moment 14-33:, equals nt+o(g,r). If, at the moment, t, the mechanism does not require energy, then the probability that it will require it at the moment 3+,r equals ght-align}, inde- pendently of the work of the other mechanisms. Set up the differential equations which P,(t), the probabilities that at the moment t, r mecha- nisms will require energy, satisfy. Find the stationary solution of these equations. 145. Two players A and B, having capital a and b, respectively, play at a game of chance consisting of separate plays. At each play, A and B have the same probability, , of winning. After each play, the loser pays 1 ruble to the winner. The game continues until the bankruptcy of one of the players. Find the probability that the second player goes bankrupt. 146. Assume that in the preceding problem the player A wins with pro- bability p}J and loses with probability q=l p. In this case what will be the probability of bankruptcy of the second player? 141* Find an integer B such that in throwing dice, the probability of the event A, that a series of three successive aces is encountered earlier than the rst series of 19 consecutive none-sea, is approximately equal to one- half. Hint. Introduce the conditional probabilities u and a ct" the event A under the conditions that the results of the rst trial are respectively ace and non-ace. Using the formula for total probability, set up the equations relating it and n. 1483\" Consider a sequence of independent trials each with three possible outcomes A , B, C of corresponding probabilities p, q and r (p 4 91+ r = 1). 33 112. Kirchhoff investigated 60 spectral lines of radiation of iron and found that each of these lines lies within 4} mm of some solar Frann hofer line. Determine whether this is \"conicidenoe\129. Let if he a nonnegative integer-valued random variable, taking the value i=0, 1, 2, with probability (A'Hcl e"). An experiment consists of choosing 1,' points independently of one another on the segment [0.1]. Denote bya-t the number of points falling on the interval {{i \"in, tin), i- l. 2, ..., n. Prove that for i=1: the JrI are independent. 130. Assume that a certain insect lays it eggs with probability (it!) e' '3 and that the probability of the evolution of an insect from an egg is ,9. Assuming the mutual independence of the evolution of the eggs. nd the probability that an insect will have exactly lo'spring. 131. In experiment \"II, M mutually exclusive outcomes A, are pamible, and in experiment .9, N mutually exclusive outcomes 3. are possible. Show that the conditional probability P(B,, I A.) can be expressed in torms of the probabilities P{A,,, | 3,} and P{B,,} in the following way: P A Pialsnn N { .IB.} \"33. ,2, P n. | 3.} Piss This relation is known as Bayes'fonnula. 131. From an urn in which there were m3}! white halls and it black balls, one hell of unknown color was lost. In order to determine the composi- tion of the balls in the urn, two balls were taken out of it at random. Find the probability that a white ball was lost if it is known that the balls taken out turned out to be white. Hint. Use the formula of Problem 13] . 133. Before certain experiments are performed. the probabilities of the mutually exclusive-and exhaustive hypotheses A . , Ag, ..., A; are considered to be n1, n2, ..., at. According to hypothesis Ah the probability of occur- rence of the event 3 at any particular realization of the experiment is en. It is known that for n. independent trials. the event it occurred m, times. It is also known that in the subsequent series of in; trials, the event 3 3-0 The probability of the sum of events occurred my times. Prove the following property of Bayes' formula: the a posterior-:7 probabilities of the hypotheses, calculated after the second series of trials and taking into account the probabilities of these hypo- theses aer the rst series of trials, are equal to the a posteriori probabili- lities. reticulated simply an the basis of the series ofn, +11: trials, in which the event Boccurred m, + in, times. 134. On a communications channel. one of three sequences of letters can betransmitted: AAAA, BBBB, CCCC, where the u prim-i probabilities of the sequences are 0.3, 0.4, 0.3, respectively. It is known that the action of noise on the receiver decreases the probability of a correct reception of a transmitted letter to 0.6. The probability of the [incorrect] reception of a transmitted letter as either of the two other letters increases to 0.2. It is assumed that the letters are distorted independently of one another. Find the probability that the sequence AAAA was transmitted if ABCA is received on the receiver. 149. Find the d.f. and the mean value of the number of tosses of a coin in Problem 93. 150. The random variables $ and n are independent, where M=2, D[$]=1, M =1, D =4. Find the expectation and dispersion of: a) $1=4-2n; b) 52=25-n. 151. Assume that in a lake there were 15,000 fish, 1000 of them marked [with radioactive tracers] (see Problem 100). 150 fish were fished out of the lake. Find the expectation of the number of marked fish among the fish caught. 152. Find the expectation and the dispersion of the number of short fibers among the randomly selected fibers in Problem 94. 153. In throwing n dice, determine the expectation, the dispersion, and central moment of the 3-rd order of the sum of the eyes on all dice. 154. Find the expectation and dispersion of the magnitude of the free motion of the molecule described in Problem 141. 155. The owner of a railway season ticket usually departs from home 39 Random variables and their properties between 7:30 and 8:00 a.m.; the journey to the station takes from 20 to 30 min. It is assumed that the time of departure and duration of the journey are independent random variables, uniformly distributed in the corresponding intervals. There are two trains which he can ride: the first departs at 8:05 a.m. and takes 35 min.; the second departs at 8:25 a.m. and takes 30 min. Assuming that he departs on one of these trains, deter- mine the mean time of his arrival at his destination. What is the probabili- ty that he will miss both trains? 156. Find the expectation and dispersion of the number of defective parts among n parts subjected to control; see Problem 102. Find also the ex- pected number of good parts occurring between two successive defective ones. 157. In Problem 130, find the expectation of the number of descendants of the insect. 158. Two dice are thrown. Find the expectation of the sum of the scores if it is known that different sides came up. 159. The diameter of a circle is measured approximately. Assuming that its magnitude is uniformly distributed in the segment [a, b], find the distribution of the area of the circle, its mean value and dispersion. 160. The density of the distribution of the absolute value of the velocity of motion of a molecule has the form p(s) = 4 9 s2 e -as