or PHARO, is the name of a game of chance. The principal rules of pharo are as follow: The banker holds a pack consisting of fifty-two cards; he draws all the cards, one after the other, and lays them down alternately at his right hand and his left; then the ponte may at his pleasure set one or more stakes upon one or more cards, either before the banker has begun to draw the cards, or after he has drawn any number of couples. The banker wins the stake of the ponte when the card of the ponte comes out in an odd place on his right hand, but loses as much to the ponte when it comes out in an even place on his left hand. The banker wins half the ponte's stake when it happens to be twice in one couple. When the card of the ponte being but once in the stock happens to be the last, the ponte neither wins nor loses; and the card of the ponte being but twice in the stock, and the last couple containing his card twice, he then loses his whole stake. Demoivre has shown how to find the gain of the banker in any circumstance of cards remaining in the stock, and of the number of times that the ponte's card is contained in it. Of this problem he enumerates four cases, viz. when the ponte's card is once, twice, three, or four times in the stock. In the first case, the gain of the banker is $\frac{1}{n}$, $n$ being the number of cards in the stock. In the second case, his gain is

$$\frac{n-2}{n \times n-1} + \frac{2}{n \times n-1} = \frac{1}{n} + \frac{1}{n \times n-1},$$

supposing $y = \frac{1}{2}$.

In the third case, his gain is

$$\frac{3}{2 \times n-1} = \frac{3}{n \times n-1},$$

supposing $y = \frac{1}{2}$. In the fourth case, the gain of the banker, or the loss of the ponte, is

$$\frac{2n-5}{n-1 \times n-3},$$

or

$$\frac{2n-5}{2 \times n-1 \times n-3},$$

supposing $y = \frac{1}{2}$. Demoivre has calculated a table, exhibiting the gain or loss in any particular circumstance of the play; and he observes, that at this play the least disadvantage of the ponte, under the same circumstances of cards remaining in the stock, is when the card of the ponte is but twice in it, the next greater when three times, the next when once, and the greatest when four times. He has also demonstrated, that the whole gain per cent. of the banker, upon all the money that is adventured at this game, is L.2. 19s. 10d. (See De moivre's Doctrine of Chances.)