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.
