Why ?

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7 Answers

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Why ?

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Let p be the required probability

If the first move is to the right(with prob 0.5), it hits 1

If the first move is to the left, then we have to find the prob of hitting 1 before hitting 2, starting from -1, which is equal to (1-p).

So p = 0.5 + 0.5*(1-p)

p=2/3

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what is browian motion??

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A martingale approach is easier and more intuitive. Standard Brownian motion is martingale, and with stopping time it is still a martingale. Given a stopping time tao, when it first hits 1 or -2, E[Btao] = B0 = 0 = p*1 + (1-p)*(-2). Therefore, 2/3.

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let desired probability be p

if the first step is 1, then p(-2 reached after 1) = 1

if the first step is -1, then if second stop is -1 , we get to -2 without getting to 1

if the first step is -1, then if second step is 1, we get back to origin

hence: p = 0.5*1 + 0.5*0.5*p + 0.5*0.5*0

p = 2/3

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a stopped martingale is still a martingale.

Ans: 2/3

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