Asking in conjuction with the previous question:
Subset Probability to Element Probability
If John selects any sized-subset (from 1 element to N elements), which is the probability of selecting every single item???
For example: Consider items 1,2,3,4. He may select {1}, {2}, {3}, {4}, {1,2}, .... {1,2,3}, {1,2,4}, ..., {1,2,3,4}.
What is the probability of his selection includes item1???
There are $2^N$ different outcomes, of which ${N-1 \choose 0} + {N-1 \choose 1} + \cdots + {N-1 \choose N-1} = 2^{N-1}$ contain the first item.