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134. On a communications channel, one of three sequences of letters can be transmitted: AAAA, BBBB, CCCC, where the a priori 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. 2.5 The probability of the sum of events 135. The events A; (i=1, 2, ..., n) are independent and P{A} =Px. Find the probability : a) of the occurrence of at least one of these events; b) the occurrence of only one of them. 136. Let A1, A2, ..., An, be random events. Prove the formulas: a ) PJUAN = [ PLAN - [ PLAInA)+ 15x P(AinA , nAN) + .. + ( - 1)tip on Ais: DPInAN = E PLAN - "I E PLAQUAN+ *=1 j = * + 1 "EI [ P(AQUA, UA, } -... + ( - 1)" - 1P) [. AN) . A=1 j=*+ 1 1=7+ 137. A schoolboy, wishing to play a trick on his friends, gathered all the caps in the cloakroom and then he hung them in a random order. What is the probability P., that at least one cap landed in its previous place, if 31 Application of the basic formulas there were altogether n hooks in the cloak-room and n caps on them? Find lim,- a Pm. 138. In an urn there are n tickets with numbers from 1 to n. The tickets are taken out randomly one at a time (without returning). What is the probability that: a) for at least one drawing, the number of the selected ticket coincides with the number of the trial being performed ; b) If m tickets are drawn (m <>