I have a question regarding two events that take place randomly over a certain time period and meet. A similar problem would be phrased as follows, which is maybe simpler:

Person A and person B each have 20 balls labelled 1 to 20. Person A picks n number of balls and person B picks m number of balls. What is the probability that at least one number is picked by both persons? What is the probability that at least two numbers are picked by both players? What is the probability that exactly two numbers are picked by both players?

I have been thinking about this and made some progress in solving it but there is an error somewhere in my solution. Can someone please help me solve this or direct me to a reference where I will be able to solve this.

Thank you very much in advance for you help

Behrouz