Question

The probability of n independent event is to ${p_1},{p_2},{p_3}, - - - - - ,{p_n}$ Find an expression for the probability that at least one of the events will happen.
A) $1 - ({p_1})({p_2})({p_3}) - - - - - ({p_n})$
B) $(1 - {p_1})(1 - {p_2})(1 - {p_3}) - - - - - (1 - {p_n})$
C) $1 - (1 - {p_1})(1 - {p_2})(1 - {p_3}) - - - - - (1 - {p_n})$
D) $({p_1})({p_2})({p_3}) - - - - - ({p_n})$

Answer 1

In the terms of probability two events are independent if the occurrence of one does not have any effect on the occurrence of second events probability. For example: Suppose there are two dies of different colors one is red and another one is black. So the probability of throwing a red die will not have any effect on the probability of a black die and vice-versa.
But if the probability of one event is dependent on the occurrence of the second event then it will be said to be dependent events. For example, there are 1 die then its outcome will have some impact on the outcome of another event.
Let the Probability for the occurrence of Independent Events = P --(1)
The probability for not occurrence of independent Events (q) = 1- p --(2)

Complete step-by-step answer:
(Step 1)Let E denote the event, p is the corresponding probability of occurrence of events, q denotes the probability of not occurrence of events.
(Step 2) Now just arrange the given information from the question.
$\Rightarrow$Probability of occurrence of $1^{st}$ event (E1) =p1
$\Rightarrow$Probability of occurrence of $2^{nd}$ event (E2) =p2
$\Rightarrow$Probability of occurrence of $3^{rd}$ event (E3) =p3
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$\Rightarrow$Probability of occurrence of an $n^{th}$ event (En) = pn
$\Rightarrow$In the same way, the probability of non-occurrence for every event can be written as
$\Rightarrow$Probability of non-occurrence of $1^{st}$ event (E1) =q1
$\Rightarrow$Probability of non-occurrence of $2^{nd}$ event (E2) =q2
$\Rightarrow$Probability of non- occurrence of $3^{rd}$ event (E3) =q3
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$\Rightarrow$Probability of non-occurrence of an $n^{th}$ event ( En) = qn
(Step 3) Now adjust the data according to the demand of the question, as in question it was asked to find the probability of occurrence of at least one event which simply means the occurrence of a minimum of 1 event ( or Occurrence of event$\geqslant$ 1).
(Step 4) One of the simple and most concept to be used is
p (At least one event) = 1- p (None occurrence) --(3)
After applying the above concept we will get the desired answer to this problem.
$\Rightarrow$ $p$(At least one event) = $1 - ({q_1})({q_2})({q_3})({q_4})({q_5}) - - - - ({q_n})$ --(4)
Now, use equation 2 and substitute it in equation 4
$\Rightarrow$ $p$(At least one event) = $1 - (1 - {p_1})(1 - {p_2})(1 - {p_3}) - - - - - (1 - {p_n})$ --(5)

The probability of occurrence of at least one event is option C.

Note: One way is to use the formula but an alternative approach to solving this question is an analytical approach by simply attempting it through manual concept. If greater than equal to 1 there simply means any event greater than or equal to one (or $1$- non-occurrence of events (which is less than one)).