A Selection of Problems from A.A. Markov’s Calculus of Probabilities: Problem 2 – A Lottery Game

Author(s):

Alan Levine (Franklin and Marshall College)

This problem is similar to the first, but Markov included an application to a lottery game dating back to 17th-century France. Games of chance were the primary motivation for some of the early writings on probability by Fermat, Pascal, Huygens, and others. Lotteries of this type are found today in many states in the US. It would be interesting to compare the probabilities and payoffs in current games to the ones mentioned here.

Задача 2^ая. Из сосуда, содержащего \(n\) билетов с нумерами \(1, 2, 3, \dots, n\) и никаких других, вынимают одновременно или последовательно m билетов, при чем, в сдучае последовательного вынимания, ни один из вынутых билетов не возвращают обратно в сосуд и новых туда также не покладывают.

Требуется определить вероятность, что между нумерами вынутых билетов появятся \(i\) нумеров, указанных заранее, напр. \(1, 2, 3, \dots, i\).

2^nd Problem. From a vessel containing \(n\) tickets with numbers \(1, 2, 3, \dots, n\), and no others, we select, simultaneously or consecutively, \(m\) tickets, so that in the case of consecutive selection, none of the selected tickets is returned to the vessel and a new one is removed.

It is required to find the probability that, among the numbers on the chosen tickets, \(i\) numbers, indicated in advance, appear—for example, \(1, 2, 3, \dots, i\).

Continue to Markov's solution of Problem 2.

Skip to statement of Problem 3.

Alan Levine (Franklin and Marshall College), "A Selection of Problems from A.A. Markov’s Calculus of Probabilities: Problem 2 – A Lottery Game," Convergence (November 2023)

Convergence

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History of Mathematics