Tuesday, September 14, 2021

[hmechntk] non-overlapping selection probability

set A: randomly draw N balls from an urn.  set B: reset the balls in the urn (if necessary), then randomly draw N balls again.

what is the probability that A and B have no elements in common?

for each set, consider drawing with or without replacement.  of course, replace all the balls when done sampling set A.

two travelers teleport randomly around the universe (inspired by Quantum Leap).  at each location, they look for evidence that the other traveler has been there, and they leave permanent evidence that they themselves were there.  what is the probability that neither encounters evidence of the other, after N random teleports?

if both sets are drawn without replacement, the answer is the hypergeometric distribution.

if both sets are drawn with replacement, the answer can probably be computed using standard tricks with the binomial distribution.  for 1 urn, the probability of a ball being selected is 1 minus the probability of it not being selected.  the probability of it not being selected in N draws with replacement is (1-1/U)^N where U is the number of balls in the urn.  2 urns require more thought, probably not too difficult.

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