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TYI 26: Attaining the Maximum

(Thanks, Dani!) Given a random sequence Image may be NSFW.
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a_1,a_2,\dots , a_n
, ***Image may be NSFW.
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a_i \in \{-1,1\}
***, Image may be NSFW.
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n>2
, let Image may be NSFW.
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S_k=a_1+a_2+\cdots +a_k
. and assume that Image may be NSFW.
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S_n=0
.  What is the probability that the maximum value of Image may be NSFW.
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S_k
is attained only for a single value of Image may be NSFW.
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k
?

Test your intuition: is this probability bounded away from 0? tends to 0 like Image may be NSFW.
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1/\sqrt n
? Quicker? Slower? Is there a nice formula?


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