Question

The Game of Pit is really fun because there are no turns. People shout out bids at random, chaotically. Here is a slightly simplified version of the game:

There are four suits, Wheat, Barley, Corn and Rye, with nine cards each, 36 cards in all. There are four players. At the opening, the cards are all dealt out, nine to each player. The players hide their cards from each other’s sight.

Players then start trading. In computer science terms, trading is asynchronous, no turns; a player can bid at any time. The only rule is that a trade must be homogeneous in suit, e.g. all Rye. (The player trading Rye need not trade all the Rye he has, though.) The player bids by shouting out the number she wants to trade, say “2!” If another player wants to trade two cards (again, homogeneous in suit), she yells out, “OK, 2!” and they trade. When one player acquires all nine of a suit, he shouts “Corner!”

Consider the situation at the time the cards have just been dealt. Imagine that you are one of the players, and Jane is another. Find the following probabilities:

(a) P(you have no Wheats). (b) P(you have seven Wheats).

(c) P(Jane has two Wheats — you have seven Wheats).
(d) P(you have a corner) (note: someone else might too; whoever shouts it out first wins).

Answer 1

In all cases, the total number of ways n(S) to deal out 9 cards to a player is ³⁶C₉

(a) If I have no Wheats, then I have been dealt any 9 out of the remaining 27 cards. Hence,

Hence, the probability of such an outcome is

(b) I have been dealt exactly 7 out of the 9 Wheats, and exactly 2 out of the remaining 27 cards. Hence,

Hence, the probability of such an outcome is

(c) The event that Jane has two Weats, given that I have seven Wheats, will occur if Jane gets the remaning two Wheats, and any 7 out of the remaining 25 cards (After I have been dealt 7 cards, there are only 27 cards remaining). Hence, total ways in which this joint event can occur is

Note that in this case, the total ways are not ³⁶C₉ , since we are also considering the ways in which Jane can be dealt 9 cards. In this case, the total ways are

Hence, the probability of such an outcome is

(d) There are only 4 ways in which I can have a corner, viz. when I have been dealt either all 9 Wheat, Barley, Corn or Rye. Hence, probability of such an outcome is