Statement-1: Let n ≥ 3 and $A_1, A_2, ....A-n$ be n independent events such that $P(A_k)=\frac{1}{k+1}$ for 1 ≤ k ≤ n, then $P(\overline{A_1}∩ \overline{A_2}∩ \overline{A_3}∩ ......∩ \overline{A_n})=\frac{1}{n+1}$
Statement -2: Let $A_1, A_2, A_3, ........., A_n $ be n(≥ 3) events associated to a random experiment. Then $A_1, A_2, A_3, ........., A-n $ are independent if $(A_1 ∩ A_2 ∩ ...........∩ A_n) = P(A_1)P(A_2)...P(A_n)$
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Answer & explanation
Correct answer: option 3
We know that three or more events associated to a random experiment are independent iff they are pairwise independent as well as as independent together. So, statement-2 is not true.
If $P(A_k)=\frac{1}{k+1}$ for $ k=1,2,.....,n$ and $ A_1, A_2, ......A_n$ are independent, then
$P(\overline{A_1}∩ \overline{A_2}∩ ......∩ \overline{A_n})= P(\overline{A_1}P(\overline{A_2})....P(\overline{A_n})$
$⇒ P(\overline{A_1}∩ \overline{A_2}∩ ......∩ \overline{A_n})=\frac{1}{2}×\frac{2}{3}×\frac{3}{4}×....×\frac{n}{n+1}=\frac{1}{n+}$
So, statement-2 is true.