Pascal-like Triangles Made from a Game

The triangle of fractions has properties that makes it very similar to Pascal's triangle: suppose two adjacent fractions in the same row are and . Then the fraction below them is , which is how the fractions and are added in the Farey sequence.

Let , , be fixed natural numbers such that . There are players seated in a circle. The game begins with the first player. Proceeding in order, a box is passed from hand to hand. The box contains red cards and white cards. When a player gets the box, he draws a card from it. Once a card is drawn, it is not returned to the box. If a player draws a red card, he loses and the game ends. Let be the probability that the player loses the game. Then for fixed numbers and with , the numbers form a Pascal-like triangle.