Probability involving the maximum of i.i.d. uniform r.v.'s

Question

The question is :

$100$ numbers are independently and uniformly distributed on $(0,1)$.Then what is the probability that the maximum of these numbers will be at most $0.9$?

How can I solve it? Please give me a hint. Then I will retry it. Thank you in advance.

Answer 1

In this answer, I made the (natural) assumption that the 100 random variable are not only identically distributed, but also independent.

Hint: look at the cumulative distribution function of $X\stackrel{\rm def}{=} \max_{1\leq k\leq n} X_k$ ($n=100$ here), and use independence of the $X_k$'s to factor. (Spoiler below.)

Spoiler #1

$X \leq x$ iff $X_k \leq x$ for all $k$.

Spoiler #2

By definition (and then independence) of the $X_k\sim\mathrm{U}(0,1)$, $$\begin{align}F_X(0.9) &= \mathbb{P}\{X \leq 0.9\} = \mathbb{P}\{\forall k, X_k \leq 0.9\} = \prod_{k=1}^n\mathbb{P}\{X_k \leq 0.9\} = \mathbb{P}\{X_1 \leq 0.9\}^n\\&= 0.9^n\end{align}$$