Question

Let $E_1$, $E_2$, $E_3$, ..., $E_n$ be independent events with respective probabilities $p_1$, $p_2$, $p_3$ ..., $P_n$. Find the probability that $(i)$ none of them occurs $(ii)$ atleast one of them occurs.

(i)	(ii)
(a) $(1-p_1)(1-p_2)\cdots (1-p_n)$	$1-[(1-p_1)(1-p_2)\cdots (1-p_n)]$
(b) $(1-p_1)(1-p_2)$	$1-[(1-p_1)(1-p_2)]$
(c) $(p_1-p_2)$	$1-(p_1-p_2)$
(d) none of these

• A. a
• B. b
• C. c
• D. d

Answer 1

Answer: (A)

For each $i=1,2,3,...,n$, $P(E_i^c)=1-P(E_i)=1-p_i$.
(i) $P$ (none of the events occurs)
$=P(E_1^c\cap E_2^c\cap E_3^c\cap ....\cap E_n^c)$
$=P(E_1^c)\times P(E_2^c)\times P(E_3^c)\times ...\times P(E_n^c))$
$=(1-P(E_1))\times (1-P(E_2))\times (1-P(E_3))\times ...\times$
$(1-P(E_n))$
$=(1-p_1)(1-p_2)(1-p_3).....(1-P_n)$.
(ii) $P$(atleast one of the events occurs) $=1-P$(none of the events occurs)
$=1-\{(1-p_1)(1-p_2)(1-p_3)...(1-P_n)\}$