Oscar Sheynin has summarized Buniakovsky's work on probability. For which see "On V. Ya. Buniakovsky's work in the theory of probability," Archive for History of the Exact Sciences,  Vol. 43, Number 3, 199-223 (1991).

This paper provides a synopsis of Buniakovsky's Russian textbook The Principles of the Mathematical Theory of Probability which was published at St. Petersburg in 1846. Sheynin notes that in it, Buniakovsky attempted to simplify the book by Laplace. He does discuss the standard topics including the treatment of observations, testimony, population statistics and so on.

I include  here  translations of two papers written in French by Buniakovsky and paraphases of two written in Russian. These are

• "Determination of the probability that a randomly chosen second degree equation with integer coefficients has real roots." St. Petersbourg Ac. Sc. Mem, T. 3/1, No. 4, (1836), p. 341-351. (Paraphrased from the Russian. Use at your own risk. However, his computations have been verified with Maple.)
  
  
  
• "On the application of the analysis of probabilities to determining the approximate values of transcendental numbers." Ibid., 457-467 and No. 5, (1837), p. 517-526. (Paraphrased from the Russian. Use at your own risk. However, his computations have been verified with Maple.)
  
  This paper is devoted to Buffon's Needle Problem and is in two parts shown as separate links above. In the first, the plane is tiled with equilateral triangles and he considers the needle falling at least once on the boundary. In the second, the needle falls on a disk. In his own textbook, Markov both generalizes the first problem to include scalene triangles and observes that Buniakovsky has made an error.
   
  
• "Sur une application curieuse de l'analyse des probabilités." Mémoires de l'Académie impériale des sciences de St-Pétersbourg, T. 6/4, No. 3-4, (1850), 233-258.
  
  Here the discussion is one of estimating the total casualties in a battle given the outcome of a previously identitified subset of soldiers.
  
  
• "Sur un instrument destiné à faciliter l'application numerique de la méthode des moindres carrés." Petersb. Bull., T. 17, No. 19 (403), (1859), 289-298.
  
  Buniakovsky had a mechanical device constructed  by which he could compute the sums required to solve least squares problems. 
• " " St. Pet. Ac. Sc. Mm. (Rs.) 15 (1869) Supplement No. 4, 20 pages.