I'm studying for my probability final and the review has some questions on Brownian motion I don't understand. One is $P(1<B_1<3)$ for B a brownian motion. Is this simply the probability that a standard normal distribution is between 1 and 3? The second question is $P(B_e < B_3)$ which since both are centered at 0 I believe would simply be 1/2 by symmetry, but I'm having some trouble showing it formally. Any help would be appreciated.
2026-04-01 16:51:31.1775062291
probability a brownian motion is between two numbers
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Your first answer is correct. For the second question you can use the fact that a B.M. has stationary increments, so $B_t - B_s$ has the same distribution as $B_{t-s}.$ Without loss of generality we can assume $e<3.$ Then $$P(B_e < B_3) =P(B_e -B_e < B_3 - B_e) = P(B_0 < B_{3-e}) = P(0 < B_{3-e})= \frac{1}{2}.$$