6 Consider a gamblers ruin where a gambler wins each game wi

6. Consider a gambler\'s ruin, where a gambler wins each game with probability 0.48 and loses each game with probability 0.52. The gambler wagers $1 on each game. He begins with $1000 and will play until be has $1020 or else goes bankrupt.

(a) Find the expected amount of money that he has at the end of the game.

(b) Suppose that the gambler is allowed to borrow money and therefore can play forever. Find the expected maximum amount of money that he will ever have.

Solution

Prob for success 0.48

losing                0.52

Wagering amount on each game =1

Starting amount = 1000

He quits till he has 1020 or 0

a) Expected money gain at the end of each game

= 1(0.48)-1(0.52)

=- -0.04

Hence for n games expected money gain = -0.04n

Expected money he has at the end of the game = 1000 - 0.04n

He plays till 1000-0.04n = 1020 or 1000 -0.04 n =0

Or n is negative impossible or n = 1000/0.04 = 25000

He plays 25000 games

At the end he goes bank rupt

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b) Maximum amount o fmoney he will have is only at the beginning 1000

As expected gain each game is negative, his balance will be decreasing after every game.