Question

lf the probabilities of $n$ lndependent events, $A_1,A_2,A_3,\dots \dots 4$ are $P_1,P_2,P_3,\dots -P_n$. The probability that at least one of the events will happen

• A. $P_1{P}_2{P}_3\dots P_n$
• B. $(1-P_1)(1-P_2)(1-P_3)\dots .(1-P_n)$
• C. $1-(1-P_1)(1-P_2)(1-P_3)\dots .(1-P_n)$
• D. $1-P_1{P}_2{P}_3\dots P_n$

Answer 1

The correct option is C $1-(1-P_1)(1-P_2)(1-P_3)\dots .(1-P_n)$

$P$(at least one of the events will happen) $=1-$$P$(none of them will happen)
Since all of them are independent, therefore
$P(\overline{A_1}\cap \overline{A_2}\cap \overline{A_3}.......\overline{A_n}\cap )=P\left(\overline{A_1}\right)\ast P\left(\overline{A_2}\right)\ast .......P\left(\overline{A_n}\right)=(1-P(A_1\left)\right)\ast (1-P(A_2\left)\right)........(1-P(A_n\left)\right)=(1-P_1)\ast (1-P_2)\ast ........(1-P_n)$
Hence required $P=1-(1-P_1)\ast (1-P_2)\ast ........(1-P_n)$