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?

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?

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.

a) What is the probability that the comes up heads at least once?

b) Given that the coin comes up heads at least once, that is the probability that it comes

up tails at least once.

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).

a) Derive the recursion Pn = p(1-P

_{n-1 })+(1-p) P_{n-1},for all n≥2 b) Prove by induction , using the recursion in part (a), that Pn =(1+(1-2p)

^{n})/2