The urn contains B blue balls and R red balls. Each trial consists of drawing one random ball from the urn and observing its color. Then the ball is placed back in the urn.
What is the probability that, after N trials, the number of times a red ball is observed is K?
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The problem is ambiguously worded. The "number of red balls" is a constant, R, so it is unlikely to ever coincide with K (irrespective of the value of N). What should be found here is
"the probability that, after N trials, the number of observations of red balls is K".