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
The Esame critico was written
in response to the memoir "Disquisitiones
analyticae de novo
coniecturali," by Daniel Bernoulli.published in 1769. In that paper
Bernoulli investigates the among others the following problem:
- Lotto. Prodromo della
Nuova Enciclopedia Italiana. Siena, Pazzini Carli e Bindi (1779).
- Esame critico di un problema
probabilità del Sig. Daniele Bernoulli, e soluzione d'un altro
problema analogo al Bernulliano. Memorie di Matematica e
della Società Italian, 1 (1782). pp. 768-824.
- Giuoco del lotto.
Antologia Romana, 11. In Roma,
presso Gregorio Settari (1785) pp. 81-95.
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