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From the linked page, a minikey is a 30-character string over the base58 alphabet with the first byte fixed to 'S', so effectively 29 characters. This gives a space of $log_2(58^{29}) \approx 169.88$ bits. Assuming that SHA is a random function, the probability of the hash starting with an 0-byte after appending a ? is 1/256, so this check loses 8 bits of ...
There's really two things to consider here: Entropy. Assuming that the hash function in question maps exactly the same number $2^{n-k}$ of bit strings of length $n$ to each hash output of length $k\leq n$, then fixing $l\leq k$ bits of the hash reduces the set of possible choices for the input from $\{0,1\}^n$ to some subset $S\subseteq\{0,1\}^n$ of ...