#### Answer

a) The probability of winning the prize with one lottery ticket is $\frac{1}{15,504}$.
b) The probability of winning the prize with $100$ different lottery tickets is $\frac{25}{3876}$

#### Work Step by Step

(a)
We know that the number of ways in which $5$ different numbers can be chosen from $1$ to $20$ is given below:
$\begin{align}
& _{20}{{C}_{5}}=\frac{20!}{5!\left( 20-5 \right)!} \\
& =\frac{20!}{5!15!} \\
& =\frac{20\times 19\times 18\times 17\times 16\times 15!}{5!15!} \\
& =15504
\end{align}$
Therefore, $ n\left( S \right)=15504$
Now, there is only one way to win the lottery. Therefore, $ n\left( E \right)=1$
Then,
$\begin{align}
& P\left( E \right)=\frac{n\left( E \right)}{n\left( S \right)} \\
& =\frac{1}{15,504}
\end{align}$
Thus, the probability of winning the prize with one lottery ticket is $\frac{1}{15,504}$.
(b)
We know that the number of ways in which $5$ different numbers can be chosen from $1$ to $20$ is given below:
$\begin{align}
& _{20}{{C}_{5}}=\frac{20!}{5!\left( 20-5 \right)!} \\
& =\frac{20!}{5!15!} \\
& =\frac{20\times 19\times 18\times 17\times 16\times 15!}{5!15!} \\
& =15504
\end{align}$
Therefore, $ n\left( S \right)=15504$
There are $100$ different tickets so, the number of ways of choosing $1$ from $100$ different tickets is ${}^{100}{{C}_{1}}=100$.
Therefore, the probability of winning the prize with one lottery ticket is
$\begin{align}
& P\left( E \right)=\frac{n\left( E \right)}{n\left( S \right)} \\
& =\frac{100}{15504}
\end{align}$
Thus, the probability of winning the prize with one lottery ticket is $\frac{25}{3876}$.