Ahead at One, Behind at Two

An index fund's cumulative tracking difference versus its benchmark follows a standard Brownian motion Wt W_t with W0=0 W_0 = 0 . What is the probability that the fund is ahead at time 1 but behind at time 2 — that is, Pr⁡(W1>0, W2<0) \Pr(W_1 > 0,\ W_2 < 0) ?

Answer

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