# Gianfranceso Malfatti

b.26 September 1731
d. 9 October 1807

The Opere of Malfatti are published in two volumes by Unione Matematica Italiana in 1981. Three items warrant attention. These are
1. Lotto. Prodromo della Nuova Enciclopedia Italiana. Siena, Pazzini Carli e Bindi (1779).
2. Memorie di Matematica e Fisica della Società  Italian, 1 (1782). pp. 768-824.
3. Giuoco del lotto. Antologia Romana, 11. In Roma, presso Gregorio Settari (1785) pp. 81-95.
The Esame critico was written in response to the memoir "Disquisitiones analyticae de novo problemate coniecturali," by Daniel Bernoulli.published in 1769. In that paper Bernoulli investigates the among others the following problem:

Three urns are given of which the first contains n white tickets, the second n black and the third n red. A ticket is drawn at random from the first urn, and then deposited into the second. Next one is drawn from the second and deposited into the third. Finally one is drawn from the third and deposited into the first. The expected number of white tickets in each urn is desired after any number of such cycles of extractions and deposits.

Bernoulli replaced the urns with vessels of white, black and red fluids and derived a solution using the differential calculus. Malfatti objected on the grounds that the problem is discrete.

In his memoir, Malfatti restricted himself to the case of two urns, the first holding white balls and the second an equal number of black. Unlike the mixing of fluids, it is possible in the exchanges that the number of white balls in the first urn vary rather than continually decrease.  In fact, the urn could return to its original state. This memoir is quite painful to read. We can model the situation as a Markov process. The first urn has  n+1 states representing the number of white balls contained in it. See these notes for a verification of the numerical results.