SATHEE: Probability 3 Question 6

Probability 3 Question 6

6. 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?

(2016 Main) (a) $E_{1}$ and $E_{2}$ are independent

(b) $E_{2}$ and $E_{3}$ are independent

(d) $E_{1}, E_{2}$ and $E_{3}$ are independent

Show Answer

Solution:

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

and $\quad \begin{aligned} E _2 & ={(1,2),(2,2),(3,2),(4,2),(5,2),(6,2)} \ E _3 & ={(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),\end{aligned}$

$(3, 2), (3, 4), (3, 6), (4, 1), (4, 3), (4, 5)$ , $Extra close brace or missing open brace$

$\Rightarrow P (E_{1}) = \frac{6}{36} = \frac{1}{6}, P (E_{2}) = \frac{6}{36} = \frac{1}{6}$

and $P (E_{3}) = \frac{18}{36} = \frac{1}{2}$

Now, $P (E_{1} \cap E_{2}) = P$ (getting 4 on die $A$ and 2 on die $B$ )

$= \frac{1}{36} = P (E_{1}) \cdot P (E_{2})$

$P (E_{2} \cap E_{3}) = P$ (getting 2 on die $B$ and sum of numbers on both dice is odd)

$= \frac{3}{36} = P (E_{2}) \cdot P (E_{3})$

$P (E_{1} \cap E_{3}) = P$ (getting 4 on die $A$ and sum of numbers on both dice is odd)

$= \frac{3}{36} = P (E_{1}) \cdot P (E_{3})$

and $P (E_{1} \cap E_{2} \cap E_{3}) = P$ [getting 4 on die $A, 2$ on die $B$ and sum of numbers is odd]

$= P (impossible event) = 0$

Hence, $E_{1}, E_{2}$ and $E_{3}$ are not independent.

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Probability 3 Question 6

6. 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?

Solution:

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Probability 3 Question 6

6. 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 and E3 is the event that the sum of numbers on both dice is odd, then which of the following statements is not true?

Solution:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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6. 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?