If I have an exponential random variable with a mean of $0.5$ and we consider a sample of length $10$; How can I calculate if its minimum value is bigger than a certain probability or how can I calculate if its maximum value is bigger than a certain probability? How can I approach this?.
(By exponential variable i assume it refers to that the random variable follows an exponential distribution)
HINT
Let $X_1, \ldots, X_n$ be independent with the same distribution, say with cdf $F(x)$. Then, $$ \mathbb{P}\left[\max_{1 \le i \le n} X_i \le x\right] = \mathbb{P}[X_1 \le x, X_2 \le x, \ldots X_n \le x] = \prod_{i=1}^n \mathbb{P}[X_i \le x] = F(x)^n $$
Can you try a similar trick on the minimum of $X_i$ using $\min \ge x$?
UPDATE $$ \mathbb{P}\left[\min_{1 \le i \le n} X_i \ge x\right] = \mathbb{P}[X_1 \ge x, X_2 \ge x, \ldots X_n \ge x] = \prod_{i=1}^n \mathbb{P}[X_i \ge x] = \left(1-F(x)\right)^n $$
Now, in general, if $X \sim \mathcal{E}(a)$ then $$F_X(x) = 1 - e^{-ax} \text{ and } \mathbb{E}[X] = 1/a.$$
Since you know the mean of the distribution, you can compute $a$ and get the actual CDF, and then evaluate at $x = 0.03$...