Huygens's Appendix to De ratiociniis in aleae ludo

Quoting from an English translation of Hugens's appendix to De ratiociniis in aleae ludo [or, On Reasoning in Games of Dice], here are the five problems given:

Problem 1. A and B play together with a pair of Dice upon this Condition, That A shall win if he throws 6, and B if he throws 7; and A is to take one Throw first, and then B two Throws together, then A to take two Throws together, and so on both of them the same, till one wins. The Question is, What Proportion their Chances bear to one another? Answ. As 10355 to 12276.

Problem 2. THREE Gamesters, A, B, and C, taking 12 Counters, 4 of which are White, and 8 black, play upon these Terms: That the first of them that shall blindfold choose a white Counter shall win; and A shall have the first Choice, B the second, and C the third; and then A to begin again, and so on in their turns. What is the Proportion of their Chances?

Problem 3. A lays with B, that out of 40 cards, i.e. 10 of each different Sort, he will draw 4, so as to have one of every Sort. And the Proportion of his Chance to that of B, is found to be as 1000 to 8139.

Problem 4. HAVING chosen 12 Counters as before, 8 black and 4 white, A lays with B that he will blindfold take 7 out of them, among which there shall be 3 black ones. Quaere [Lat., "Search out"], What is the Proportion of their Chances?

Problem 5. A and B taking 12 Pieces of Money each, play with 3 Dice on this Condition, That if the Number 11 is thrown, A shall give B one Piece, but if 14 shall be thrown, then B shall give one to A; and he shall win the Game that first gets all the Pieces of Money. And the Proportion of A's Chance to B's is found to be, as 244,140,625 to 282,429,536,481.

This is also known as the "gambler's ruin" problem. Pascal first posed this problem to Fermat (and it was communicated to Huygens via Carcavi in a letter dated September 28, 1656). To be clear: A keeps tossing the dice until the game is over.

For more about the history and context of these problems, see Anders Hald's A History of Probability and Statistics and Their Applications Before 1750 pp.74–78.