# Probabilty to win in die rolling game

A and B play rolling an $N$-faced die having numbers $1,2,3..,N$. A takes the first turn. After that both players take alternative turns. Whoever gets a number which is lesser or equal to the number obtained by the opponent in the previous turn loses the game. Let $P(k)$ be the probability of losing after rolling the die if opponent obtained $k$ in the previous turn. Write a recursive expression for $P(k)$. Also find the probability that A loses the game.

My thoughts:

If my opponent obtains some number $1 \le k \le N$ in the previous turn, then $p(k)$, the probability that I lose, should be $k/N$ since favourable outcomes of a die roll are $[1,k]$ and there are obviously $N$ possible outcomes. So I wish to obtain a recursive expression for this.

Now I think $p(k+1)>p(k)$ so I think the recursive expression should be $p(k+1)=p(k)+1/N$, $1/N$ added for one more extra term $k+1$.

Where do these ideas need correction?

Thanks Hagen and Peter for the hint , i ll attempt a recursive solution P(N)=prob to lose if the opponent gets N is the previous turn Clearly P(N) = 1 P(loose) = loose in this chance + P(u dont loose).P(opponent doesnot loose).P(then u loose)+...

P(N-1)=N-1/N + (1/N).(1-P(N)).P(lose this time)=N-1/N . . P(N-2)=N-2/N + (1/N)(1-P(N-1)) + 0 please help me generalize this

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What have you tried? If A rolls 3, what is the chance that B loses immediately? If A rolls k, what is the chance that B loses immediately? – Ross Millikan Nov 6 '13 at 21:16
Hint: Try recursion "the other way". That is start with the obvious $p(N)=?$ and determine $p(k)$ from knowing all $p(i)$, $i\ge k$. – Hagen von Eitzen Nov 6 '13 at 21:21
This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. – Did Nov 6 '13 at 21:57
Thanks for editing to add your ideas. The point at which you're going wrong is to interpret the question as asking for a probability which relates to a single die roll. It's actually asking for the probability that you lose on the first die roll (which you correctly state to be $k/N$), or that you lose on your next die roll, or a subsequent one. I hope that makes it clearer what you should be recursing on. – Peter Taylor Nov 7 '13 at 12:29
Hey, can anyone tell me what exactly is the problem with his approach and also, I think there is no question of subsequent die rolls, as either of A or B has to win. Am I wrong? – Frustrated Coder Nov 10 '13 at 9:33