In a standard gambler's ruin problem with probabilities p= 0.50, find the expected duration of the game provided that the usual stakes of k units for the gambler and (a-k) units for opponent. if p=0.83 find the expected duration of game.
Answers
Answer:
Step-by-step explanation:
1 Gambler’s Ruin Problem
Let N ≥ 2 be an integer and let 1 ≤ i ≤ N − 1. Consider a gambler who starts with an
initial fortune of $i and then on each successive gamble either wins $1 or loses $1 independent
of the past with probabilities p and q = 1 − p respectively. Let Xn denote the total fortune
after the n
th gamble. The gambler’s objective is to reach a total fortune of $N, without first
getting ruined (running out of money). If the gambler succeeds, then the gambler is said to win
the game. In any case, the gambler stops playing after winning or getting ruined, whichever
happens first.
{Xn} yields a Markov chain (MC) on the state space S = {0, 1, . . . , N}. The transition
probabilities are given by Pi,i+1 = p, Pi,i−1 = q, 0 < i < N, and both 0 and N are absorbing
states, P00 = PNN = 1.1
For example, when N = 4 the transition matrix is given by
P =
1 0 0 0 0
q 0 p 0 0
0 q 0 p 0
0 0 q 0 p
0 0 0 0 1
.
While the game proceeds, this MC forms a simple random walk
Xn = i + ∆1 + · · · + ∆n, n ≥ 1, X0 = i,
where {∆n} forms an i.i.d. sequence of r.v.s. distributed as P(∆ = 1) = p, P(∆ = −1) = q =
1 − p, and represents the earnings on the successive gambles.
Since the game stops when either Xn = 0 or Xn = N, let
τi = min{n ≥ 0 : Xn ∈ {0, N}|X0 = i},
denote the time at which the game stops when X0 = i. If Xτi = N, then the gambler wins, if
Xτi = 0, then the gambler is ruined.
Let Pi(N) = P(Xτi = N) denote the probability that the gambler wins when X0 = i.
Pi(N) denotes the probability that the gambler, starting initially with $i, reaches a
total fortune of N before ruin; 1 − Pi(N) is thus the corresponding probably of ruin
Clearly P0(N) = 0 and PN (N) = 1 by definition, and we next proceed to compute Pi(N), 1 ≤
i ≤ N − 1.
Proposition 1.1 (Gambler’s Ruin Problem)
Pi(N) =
1−(
q
p
)
i
1−(
q
p
)N , if p 6= q;
i
N
, if p = q = 0.5.
(