YLL討論網

#ed_op#DIV#ed_cl#1.  In data communications ,a  message transmitted from one end is subject to various sources of distortion and may be received erroneously at the other end. Suppose that a message of 64 bits (a bit is the smallest unit of information and is either 1 or 0) is transmitted through a medium. If each bit is received incorrectly with probability 0.0001 independently of the other bits, what is the probability that the message received is free of error?#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#2.  For the sake of  argument , suppose that an aircrafe engine fails in flight with probability 0.4, independently of the plan's other engines. Also ,suppose that a plane can complete its journay successfully if at least half of its engins do not fail. Under such circumstance , would you prefer to fly on a four-engine plane or a two-engine plane?#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#3. A coin has probability of its coming up heads on a flips is 1/3. The coin is flipped 10 times. Assume that the tosses are all independent of each other.#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#    a) What is the probability that the comes up heads at least once?#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#    b) Given that the coin comes up heads at least once, that is the probability that it comes #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#        up tails at least once.#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#4. Suppose that  we have a biased coin with P(H) =p and P(T)=1-p. Given an integer n≥1, let Pn denote the probability that, in n tosses of  the coin, the total number of heads that occur is an even number (which also includes 0). #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#    a) Derive the recursion Pn = p(1-P#ed_op#SUB#ed_cl#n-1  #ed_op#/SUB#ed_cl#)+(1-p) P#ed_op#SUB#ed_cl#n-1#ed_op#/SUB#ed_cl#   ,for all n≥2#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#    b) Prove by induction , using the recursion in part (a), that  Pn =(1+(1-2p)#ed_op#SUP#ed_cl#n#ed_op#/SUP#ed_cl#)/2 #ed_op#/DIV#ed_cl#