a$.$ Recall that when there are $N$ active nodes, the efficiency of slotted ALOHA is $Np(1-p{)}^{N-1}.$ Find the value of $p$ that maximizes this expression.

b$.$ Using the value of $p$ found in $($a$),$ find the efficiency of slotted ALOHA by letting $N$ approach infinity. Hint: $(1-\frac{1}{N}{)}^N$ approaches $\frac{1}{e}$ as $N$ approaches infinity.