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 a/b and c/d. Then the fraction below them is (a+c)/(b+d), which is how the fractions a/b and c/d are added in the Farey sequence. Let p, n, m be fixed natural numbers such that m<=n. There are p 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 m red cards and n-m 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 F(p,n,m,v) be the probability that the v^(th) player loses the game. Then for fixed numbers p and v with v<=p, the numbers {F(p,n,m,v), n=1,2,...,m<=n} form a Pascal-like triangle