probability statistics

Question

A casino runs a game which consists of a sequence of coin tosses, if toss head, then gain +1 and the probability is 1/2. If toss the tail, then gain -1 and the probability is 1/2. A gambler starts with initial fortune Z. Goal is to increase total fortune to g. (think of, say, g= 100). The gambler decides to play until he reaches g or goes broke (hits zero)

Suppose you start with an initial fortune z and your goal is to reach g. what is the probability the first toss was heads, given that you reach goal?

Answer 1

It is easy to see that P(z) = z/g (If you wish, you can easily show that this solves P(0) = 0, P(g) = 1, P(z) = 1/2 P(z-1) + 1/2 P(z-1), z = 1, 2, ..., g-1)

Then, P(first toss is heads | you win) = P(you win and first toss is heads)/(P(you win and first toss is heads)+P(you win and first toss is tails)) =

P(first toss is heads)P(win|first toss is heads)/(P(first toss is heads)P(win|first toss is heads)+P(first toss is tails)P(win|first toss is tails)) = 1/2P(z+1)/(1/2P(z+1)+1/2P(z-1)) =

P(z+1)/(P(z+1)+P(z-1)) =

(z+1)/g/((z+1)/g+(z-1)/g) =

(z+1)/(2z) or 1/2 + 1/(2z)