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By Norman L. Johnson, Samuel Kotz

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**Extra info for Urn models and their application**

**Sample text**

1 in the case n = 3. ~fuat is the probability that B will play with A at the final match? 47. Let n indistinguishable particles be distributed at random into n different cells. Each cell may contain an arbitrary number of particles. 10). Determine the probability that: (a) a fixed cell will contain exactly k particles, k n; (b) exactly m cells will be empty, m n; (c) in each cell there will be at least two particles. < < ELEMENTARY PROBABILITY first round 21 second round final 2 3 winner 4 5 6 7 B Fig.

Prove the followinq. equalities: (a) P(An IAm) = PtAm IAn ); (b) PtA IA ) n m P(A IA); (c) P(A IA) = P(A IA); (d) P(A IA) = P(A IA). 14. Consider two urns and 20 balls of which ten are white and ten are black. How should these balls be distributed in the two urns by number and by colour so that if one of the urns is chosen at random and one ball is drawn from it, the probability that this ball is white is maximized? 15. In an urn there are n white balls, numbered with the integers from one to n, and n black balls, also numbered with the integers from one to n.

Show that if two events are independent and each of them has a positive probability, then these events are not mutually exclusive. Solution. We have P(AB) = P(A)P(B). , A and B are not mutually exclusive. 2. It is known that 4% of the items of a lot are defective, and 75% of the good items are first-grade. Calculate the probability that a chosen at random item will be first-grade. Solution. Let A = {the chosen item is not defective} and B = {the chosen item is a first-grade one}. We are looking for P(AB).