Here is an example of a toy Bloom filter. It uses an array of $10$ bits and two independent hash

functions $f$ and $g.$ We want to test membership in a set $S$ of three elements, so we hash each of

the three elements using both $f$ and $g,$ and we set to $1$ any bit that any of the three elements is

hashed to by either of the hash functions.

When a new element $x$ arrives, we compute $f(x)$ and $g(x),$ and we say $x$ is in the set $S$ if both $f(x)$

and $g(x)$ are $1.$ Assume $x$ is not in the set $S.$ What is the probability of a false positive, i$.$e$.,$ the

probability of saying that $x$ is in $S?$ Please explain how you get this probability.