A Markov chain on non-negative integers which arises in a roulette-type game is discussed. The transition probabilities are p01 = ρ,pNj = δNj,pi,i+W = q, pi,i−1 = p = 1 − q,1  W < N, 0  ρ  1,N− W<j N and i = 1,2,...,N− W. Using formulae for the determinant of a partitioned matrix, a closed form expression for the solution of the Markov chain roulettetype game is deduced. The present analysis is supported by two mathematical models from tumor growth and war with bargaining.