Let two fair six-faced dice A and B be thrown simultaneously. If is the event that die A shows up four

Question

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, then which of the following statements is true?

Smart Achievers · Accepted Answer

(4)

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

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

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

$P (E_{1}) = \frac{1}{6}; P (E_{2}) = \frac{1}{6}; P (E_{3}) = \frac{1}{2}$

$P (E_{1} \cap E_{2}) = \frac{1}{36}, P (E_{2} \cap E_{3}) = \frac{1}{12}, P (E_{1} \cap E_{3}) = \frac{1}{12}$

And $P (E_{1} \cap E_{2} \cap E_{3}) = 0 \neq P (E_{1}) . P (E_{2}) . P (E_{3})$

$\Rightarrow 0 \neq \frac{1}{6} . \frac{1}{6} . \frac{1}{2} \Rightarrow E_{1}, E_{2}, E_{3} are not independent.$

Solution

Q.
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, then which of the following statements is true

1 $E_{1}$ and $E_{2}$ are exclusive

2 $E_{2}$ and $E_{1}$ are exhaustive

3 $E_{1}$ and $E_{2}$ are dependent

4 $E_{1}$ , $E_{2}$ are independent

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Solution

Q. Let two fair six-faced dice A and B be thrown simultaneously. If E1 is the event that die A shows up four, E2 is the event that die B shows up two, then which of the following statements is true

1 E1 and E2 are exclusive

2 E2 and E1 are exhaustive

3 E1 and E2 are dependent