A binary communication system is used to send one of two mes

A binary communication system is used to send one of two messages: (j) message A is sent with probability 2/3, and consists of an infinite sequence of zeroes, (ii) message B is sent with probability 1/3, and consists of an infinite sequence of ones. The ith received bit is correct (i.e., the same as the transmitted bit) with probability 3/4, and is incorrect (i.e., a transmitted 0 is received as a 1, and vice versa), with probability 1/4. We assume that conditioned on any specific message sent, the received bits, denoted by Y1,Y2,... are independent. Note: Enter numerical answers; do not enter ! or combinations. 1. Find P(Y1, 0) ,the probability that the first bit received is 0. 2. Given that message A was transmitted, what s the probability that exactly 6 of the first 10 received bits are ones? (Answer with at least 3 decimal digits.) 3. Find the probability that the first and second received bits are the same. 4. Given that Y ,... Y5 were all equal to 0, what is the probability that Y6 is also zero? 5. Find the mean of K, where K = min {i :Yi =1) is the index of the first bit that is 1.

Solution

1. (2/3*3/4) + (1/3*1/4) = 7/12 = 0.583

2. [10C6*(1/4)^6*(3/4)^4] = 0.016

3. Probability of both zero
   (7/12)^2
Probability of both 1
(1 - 7/12 )^2 = (5/12)^2
Probability of both bits equal = (7/12)^2 + (5/12)^2 = 0.514

4. Since all bits are independent of each other, first 5 bits wil not affect the sixth one.
   So, the probability that 6th bit is zero is also 7/12 = 0.583

5. For i=1, p = 5/12
For i=2, p = 7/12*(5/12)
For i=n, p = (7/12)^(n-1)]*(5/12)

Summing the series n*(7/12)^(n-1)*(5/12)
We get 12/5

Hence, Mean = 12/5 = 2.4