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)
$= P(E^c_1 \cap E_2^c \cap E_3^c \cap .... \cap E_n^c)$
$= P (E^c_1) \times P (E_2^c) \times P (E_3^c) \times ... \times P(E^c_n))$
$= (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)
$= 1 - \{ ( 1 - p_1) ( 1 - p_2) ( 1 - p_3) ...( 1-P_n )\}$