# A Selection of Problems from A.A. Markov’s Calculus of Probabilities: Problem 1 – Selecting Balls from a Vessel

Author(s):
Alan Levine (Franklin and Marshall College)

This problem is very elementary and can be found in nearly every modern probability textbook, where it might be phrased as “selection without replacement.” It is a standard hypergeometric distribution example.

 Задача 1ая.  Из сосуда, содержащего $a$ белых и $b$ чёрных шаров и никаких других, вынимают одновременно или последовательно $\alpha+\beta$ шаров, при чем, в случае последовательного вынимания, ни один из вынутых шаров не возвращают обратно в сосуд и новых туда также не подкладывают. Требуется определить вероятность, что между вынутыми таким образом шарами будеть $\alpha$ белых и $\beta$ чёрных.

1st Problem. From a vessel containing $a$ white and $b$ black balls and no others, we select, simultaneously or consecutively, $\alpha + \beta$ balls, for which, in the case of consecutive selection, none of the chosen balls are returned to the vessel and new ones are also not put back in.

It is required to determine the probability that there will be $\alpha$ white and $\beta$ black among the balls selected in this way.

Figure 4. Augustus De Morgan’s illustration of a ball and vessel problem with three balls,
from page 53 of his 1838 Essay on Probabilities. Convergence Mathematical Treasures.

Continue to Markov's solution of Problem 1.