Pascal-like Triangle Made from a Card Game
Pascal-like Triangle Made from a Card Game
Let , , and be fixed positive integers, with . There are players. Each draws cards in turn from a pile of cards containing red cards and white cards. A player immediately loses the game after drawing a red card. This game is mathematically equivalent to a Russian roulette game in which players take turns.
p
m
n
s
m≤n
p
s
m
n-m
p
Let be the probability that the player loses the game. Then for fixed positive integers , and , the set of denominators and numerators of for , , have a very similar pattern to the pattern of Pascal's triangle.
f(p,m,n,s,v)
th
v
p
s
v
f(p,m,n,s,v)
m=n
n=1,2,…