The player starts with £200, with the aim to get £1000. He only takes 50/50 bets. If he loses, he loses all of his stake. If he wins, his return is double what he staked (his net profit is the stake). If he has less than £500, he gambles all his money, if he has more than £500, he only gambles the amount that would take him to £1000, if he wins.
Find the probability he succeeds in reaching £1000. Find the expected amount of gambles that take place in the game.
I understand to use Markov and hitting times in this, but I am struggling to start off. When drawing a diagram, I struggle as the condition that he only places what he needs when he is wealthier than £500 means that the graph is impossible to draw.. Any help would be greatly appreciated!
Best Answer
From £$200$, he can go to £$400$ or £$0$
From £$400$, he can go to £$800$ or £$0$
From £$800$, he can go to £$1000$ or £$600$
From £$600$, he can go to £$1000$ or £$200$
This gives all possible states of the Markov chain.