Probability of an aleatory experiment conducted $n$ times

Question

What is the probability of an aleatory experiment conducted $n$ times? For example, say we choose randomly a number $x$ from a known interval, which happens to contain some certain kind of numbers we're going to call Specifics. So, if we know that the size of the set containing all the Specifics is half the size of the entire interval, the probability must be: $$P(S) = \frac{N(S)}{N(Ω)} = \frac{1}{2}$$ So, if we choose to go with $n$ different random $x$-es, is the probability $P(S)^n$? If so, why?

Answer 1

Let $p \equiv P(S)$. The probability of all $n$ draws satisfying the condition is, in fact, $p^n$.

The probability of exactly $k$ out of the $n$ draws satisfying the condition is given by

$$\binom n k p^k(1-p)^{n-k}$$

In particular, the probability of exactly $1$ success is given by

$$ \binom n 1 p(1-p)^{n-1}$$

Finally the probability of at least one success is equal $1$ minus the probability of missing every draw,

$$1-p^n $$