Probability of choosing a certain random number

Question

Given person A chose a number out of the infinity set N, does the probability of person B choosing the same number defined?

Answer 1

There is no way to choose a random number out of $\Bbb N$ such that all the numbers have equal chance to be chosen. You can choose a random number with a probability distribution such that the sum of all the probabilities is $1$. One such is $P(n)=2^{-n}$ (and $P(0)=0$ if you think $0 \in \Bbb N$). If you know the probability distributions used by $A$ and $B$, the probability they pick the same number is well defined. If $A$ uses the probability distribution $P_A(n)$ and $B$ uses $P_B*n)$ the chance they choose the same number is $\sum_{n=1}^\infty P_A(n)P_B(n)$. This is the same sum you would use for finite sets. As the sum is monotonically increasing and bounded above by $1$, it converges nicely.