Find probability of B

Question

Computer A produced a bit. This bit was then sent to computer B, then from B sent to computer C, and finally from C to computer D. Every time one computer sends a bit to another, there's a 56% chance that an error will occur and the bit will change (from 0 to 1, or from 1 to 0). If the bit arrived "successfully" (meaning the value of the bit at A is the same as at D, an even number (including no changes at all) of changes occurred) to D, what's the probability that the bit was not changed when sent from A to B?

Answer 1

Let $0$ denote no change from one computer to the next, and let $1$ denote a change. Then the four different ways a bit can arrive successfully is $000$ with a probability of $ 0.44^3 $ and $110$, $101$ and $011$, each with a probability of $ 0.44\cdot 0.56^2 $. Given that the bit arrived successfully, the probability for a change from computer A to computer B is $$ \frac{0.44^3+0.44\cdot 0.56^2}{0.44^3+3\cdot0.44\cdot 0.56^2}. $$

Answer 2

This seems like a standard conditional probability problem. It is just the probability of no change divided by the probability of an even number of changes; the sum of the probabilities of $0$ and $2$ changes. The probabilities are each binomial.

If you have more problems with this, show some work so we can tell where the problem is.