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  • Two dice are tossed. Find whether the following two events A and B are independent: A = {(x, y): x+y = 11} and B = {(x, y): x ≠ 5} where (x, y) denotes a typical sample point.

Two dice are tossed. Find whether the following two events A and B are independent:
A = {(x, y): x+y = 11} and B = {(x, y): x ≠ 5}
where (x, y) denotes a typical sample point.

Answers (1)

Given-

A= {(x, y):x+y=11}

And B= {(x, y): x≠5}
∴ A = {(5,6), (6,5)}

B= {(1,1), (1,2), (1,3), (1,4), ((1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

\begin{aligned} &\Rightarrow \mathrm{n}(\mathrm{A})=2, \mathrm{n}(\mathrm{B})=30, \mathrm{n}(\mathrm{A} \cap \mathrm{B})=1\\ &P(A)=\frac{2}{36}=\frac{1}{18} \text { And } \mathrm{P}(\mathrm{B})=\frac{30}{36}=\frac{5}{6}\\ &\Rightarrow P(A) \cdot P(B)=\frac{5}{108} \text { And } P(A \cap B)=\frac{1}{18} \neq P(A) \cdot P(B)\\ &\text { Hence, } \mathrm{A} \text { and } \mathrm{B} \text { are not independent. } \end{aligned}

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infoexpert22

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