Q$2.$ Consider the Bloom filter. Define $k=$ number of hash functions; $N=$ number of bits in hash table; and $D=$ number of words in dictionary. $($a$)$ Show that the expected number of bits in the hash table that are equal to zero is expressed as

$=(1-\frac{k}{N}{)}^D$

$($b$)$ Show that the probability that an input word, not in the dictionary, will be falsely accepted as being in the dictionary is

$P=(1-{)}^k$

$($c$)$ Show that the preceding expression can be approximated by

$P$~~$(1-e^{-k\frac{D}{N}}{)}^k$