An array of n bits, again

I love simple problems on simplest data structure: the infamous array of N bits. The linear scan of the array is O(N) and everyone knows this. Here I am asking: what is the probability of finding in position K-th the first bit set to 1 and all the preceding bits set to 0?

Now, Suppose that two sources are sending a stream of packets on the same wire and that they are able to detect collisions. After K failures, they suspend the transmission for random number of milliseconds in [0, 2^K-1]. What is the probability of having another failure?