SATHEE: Probability 3 Question 22

Probability 3 Question 22

22. If $X$ and $Y$ are two events such that $P (X / Y) = \frac{1}{2}, P (Y / X) = \frac{1}{3}$ and $P (X \cap Y) \frac{1}{6}$ . Then, which of the following is/are correct?

(2012)

(a) $P (X \cup Y) = 2 / 3$

(b) $X$ and $Y$ are independent

(d) $P (X^{c} \cap Y) = 1 / 3$

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Solution:

PLAN

(i) Conditional probability, i.e. $P (A / B) = \frac{P (A \cap B)}{P (B)}$

(ii) $P (A \cup B) = P (A) + P (B) - P (A \cap B)$

(iii) Independent event, then $P (A \cap B) = P (A) \cdot P (B)$

Here, $P (X / Y) = \frac{1}{2}, P \frac{Y}{X} = \frac{1}{3}$

and $P (X \cap Y) = 6$

$∴ P \frac{X}{Y} = \frac{P (X \cap Y)}{P (Y)}$

$\Rightarrow \frac{1}{2} = \frac{1 / 6}{P (Y)} \Rightarrow P (Y) = \frac{1}{3}$

$P \frac{Y}{X} = \frac{1}{3} \Rightarrow \frac{P (X \cap Y)}{P (X)} = \frac{1}{3}$

$\Rightarrow \frac{1}{6} = \frac{1}{3} P (X)$

$∴ P (X) = \frac{1}{2}$

$P (X \cup Y) = P (X) + P (Y) - P (X \cap Y)$

$= \frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{2}{3}$

$P (X \cap Y) = \frac{1}{6}$ and $P (X) \cdot P (Y) = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6}$

$\Rightarrow P (X \cap Y) = P (X) \cdot P (Y)$

i.e. independent events

$∴ P (X^{c} \cap Y) = P (Y) - P (X \cap Y)$

$= \frac{1}{3} - \frac{1}{6} = \frac{1}{6}$

पिछला

अगला

Probability 3 Question 22

22. If $X$ and $Y$ are two events such that $P (X / Y) = \frac{1}{2}, P (Y / X) = \frac{1}{3}$ and $P (X \cap Y) \frac{1}{6}$ . Then, which of the following is/are correct?

Solution:

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एनसीईआरटी पुस्तकें और समाधान

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अन्य संसाधन

पाठ्यक्रम

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नमूना मॉक टेस्ट

सदस्यता

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Probability 3 Question 22

22. If X and Y are two events such that P(X/Y)=12,P(Y/X)=13 and P(X∩Y)16. Then, which of the following is/are correct?

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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22. If $X$ and $Y$ are two events such that $P (X / Y) = \frac{1}{2}, P (Y / X) = \frac{1}{3}$ and $P (X \cap Y) \frac{1}{6}$ . Then, which of the following is/are correct?