A Bloom Filter consists of $m$ bits of memory and $h$ uniform and independent hash functions

$f_1,f_2,$dots,$f_h.$ Each $f_i$ hashes a key $k$ to an integer in the range $1,m.$ Initially all $m$ filter bits are

zero, and the data set is empty. When key $k$ is added to the data set, bits $f_1(k),f_2(k),$dots,$f_h(k)$ of the

filter are set to $1.$ When a query $"$Is key $k$ in the data set?" is made, bits $f_1(k),f_2(k),$dots,$f_h(k)$ are

examined. The query answer is "maybe" if all these bits are $1.$ Otherwise, the answer is $"$no$."$ A

filter error occurs whenever the answer is "maybe" and the key is not in the data set. Assume that

key $k$ is an integer in the range $1,n$ and $u$ updates are made. Compute the probability of filter error

for an arbitrary query after the u$-$th update. $(10\%)$