The probabilities of happening of three independent events are $p_1,p_2\text{ and }{p}_3$, then the probability of happening atleast one of them, is

Let A, Band C be three events such that, 

$P\left(A\right)=p_1,P\left(B\right)=p_2,P\left(C\right)=p_3$

Now,

$\begin{array}{r}P\left(\overline{A}\right)=1-P\left(A\right)=\left(1-P_1\right)\\ P\left(\overline{B}\right)=1-P\left(B\right)=\left(1-P_2\right)\\ P\left(\overline{C}\right)=1-P\left(C\right)=\left(1-P_3\right)\end{array}$

Probability of that happening of A, Band C

$\begin{array}{r}P(\overline{A}\cap \overline{B}\cap \overline{C})=P\left(\overline{A}\right)\cdot P\left(\overline{B}\right)\cdot P\left(\overline{C}\right)\\ =\left(1-p_1\right)\left(1-p_2\right)\left(1-p_3\right)\end{array}$

$\therefore$ Probability of happening of atleast one of them

$=1-\left(1-p_1\right)\left(1-p_2\right)\left(1-p_3\right)$