Question
How many independent $\mathrm{Uniform}(0,1)$ random variables must we generate to ensure that with probability at least $95%$ at least one of them lies in the interval $[0.7, 0.72]$?
Solution
Let $E_i$ denote the event $U_i \in [0.7, 0.72]$.
Each has probability
$$
\mathbb{P}(E_i) = 0.02.
$$
We want $$ \mathbb{P}!\left( \bigcup_{i=1}^N E_i \right) \ge 0.95. $$
Since the $U_i$ are independent, $$ \mathbb{P}!\left( \bigcup_{i=1}^N E_i \right) = 1 - \mathbb{P}!\left( \bigcap_{i=1}^N E_i^c \right) = 1 - (0.98)^N. $$
Thus we need $$ 1 - 0.98^N \ge 0.95 \quad\Longleftrightarrow\quad 0.98^N \le 0.05. $$
Taking logarithms, $$ N \log(0.98) \le \log(0.05) \quad\Longrightarrow\quad N \ge \frac{\log 0.05}{\log 0.98}. $$