Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_{1}$ is the event that die $A$ shows up four, $E_{2}$ is the event that die $B$ shows up two $E_{3}$ is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true?

E_{2}

and

E_{3}

are independent

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E_{1}

and

E_{3}

are independent

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E_{1}, E_{2}

and

E_{3}

are independent

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E_{1}

and

E_{2}

are independent

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Solution

The correct option is D $E_{1}$ and $E_{2}$ are independent
$\begin{matrix} P (E_{1}) = \frac{1}{6} P (E_{2}) = \frac{1}{6} P (E_{3}) = \frac{1}{2} P (E_{1} \cap E_{2}) = \frac{1}{36} P (E_{2} \cap E_{3}) = \frac{1}{12} P (E_{3} \cap E_{1}) = \frac{1}{12} P (E_{1} \cap E_{2} \cap E_{3}) = 0 ∴ E_{1}, E_{2}, E_{3} a r e n o t i n d e p e n d e n t . \end{matrix}$

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Let two fair six - faced dice A and B be thrown simultaneously. If $E_{1}$ is the event that die A shows up four, $E_{2}$ is the event that die B shows up two and $E_{3}$ is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true?

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