O.B. Sheynin has given in the Archive for the History of the Exact Sciences Vol. 7, pp. 246- 256 a summary of Lambert's work on probability. This he divides into three parts: theory of probability, demographical statitistics, and theory of errors. His outline is followed below.
In "Examen d'une espece de superstition ramenée au calcul des probabilités" published in Nouveau Mémoires Acad. Roy. Sci. et Belles Lettres Berlin 1771 (1773) he consideres derangements. This is the game of Treize or Rencontre studied by Nicolas Bernoulli, Moivre, Euler and Laplace.
His thoughts on subjective probability may be found in the Neues Organon (1764) Band I and Band II. In the former is "Alethiologie oder Lehre von der Wahrheit," pages 453-592 and in the latter is "Phänomenologie oder Lehre von dem Schein," pages 217-435. That is, these are the "Theory of the Truth" and the "Theory of Appearance."
Published in the Opera mathematica of J. H. Lambert is "Mathematische Ergötzungen über die Glücksspiele." This appeared in 1799 long after his death and consists of three notes on games of chance: the card game known as the lottery, dicing and heads & tails.
Lambert studies mortality, life expectancy, duration of marriages among other related topics in "Anmerkungen über die Sterblichkeit, Todtenlisten, Geburthen und Ehen." found in Theil 3 1772 pp. 476-569. See also his correspondence in Lamberts deutscher Gelehrter Briefwechsel Bd 4 pp 365-368.
Daniel Bernoulli also studied the duration of marriages.
The Photometria, sive, De mensura et gradibus luminis, colorum et umbrae (1760) : Parts I and II, Parts III, IV and V, Parts VI and VII is the primary source for understanding Lambert's theory of errors. An English translation by David L. DiLaura is available as Photometry, or, On the measure and gradations of light, colors, and shade (2001). The relevant sections are 271-306. We find also in "Anmerkungen und Zus tze zur practischen Geometrie," Part I, pp. 1-313 of the Beyträge zum Gebrauche der Mathematik und deren Anwendung I, a discussion of the errors. This is followed later by "Theorie der Zuverlässigkeit der Beobachtungen und Versuche" pp. 424-487.Additional sources: See also the site maintained by Maarten Bullynck of the works of Lambert.