Let two fair six-faced dice $A$ and $B$ are 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?

E_{1}

and

E_{2}

are independent

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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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Solution

The correct option is D $E_{1}, E_{2}$ and $E_{3}$ are independent
$P (E_{1}) = \frac{1}{6}$
$P (E_{2}) = \frac{1}{6}$

$P (E_{3}) = \frac{2 + 4 + 6 + 4 + 2}{36} = \frac{1}{2}$
$P (E_{2} \cap E_{3}) = \frac{1}{2} \times \frac{1}{6} = P (E_{2}) \times P (E_{3})$

$P (E_{1} \cap E_{3}) = \frac{1}{2} \times \frac{1}{6} = P (E_{1}) \times P (E_{3})$
$P (E_{1} \cap E_{2} \cap E_{3}) = 0$
$P (E_{1}) P (E_{2}) P (E_{3}) \neq 0$
$∴ P (E_{1} \cap E_{2} \cap E_{3}) \neq P (E_{1}) P (E_{2}) P (E_{3})$

$E_{1}, E_{3}$ are independent
$E_{2}, E_{3}$ are independent.

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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?

Let two fair six-faced dice A and B be thrown simulatneously. 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/are true ?