SATHEE: Probability 3 Question 24

Probability 3 Question 24

24. The probabilities that a student passes in Mathematics, Physics and Chemistry are $m, p$ and $c$ , respectively. Of these subjects, the students has a $75$ chance of passing in atleast one, a $50$ chance of passing in atleast two and a $40$ chance of passing in exactly two. Which of the following relations are true?

(a) $p + m + c = \frac{19}{20}$

(b) $p + m + c = \frac{27}{20}$

(d) $p m c = \frac{1}{4}$

$(1999, 3 M) (2011)$

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Solution:

Let $A, B$ and $C$ respectively denote the events that the student passes in Maths, Physics and Chemistry.

It is given,

$P (A) = m, P (B) = p$ and $P (C) = c$ and

$P$ (passing atleast in one subject)

$= P (A \cup B \cup C) = 0.75$

$\Rightarrow 1 - P (A^{'} \cap B^{'} \cap C^{'}) = 0.75$

$∵ [P (A) = 1 - P (\bar{A})$

and $[P (\overset{―}{A \cup B \cup C}] = P (A^{'} \cap B \cap C^{'})]$

$\Rightarrow 1 - P (A^{'}) \cdot P (B^{'}) \cdot P (C^{'}) = 0.75$

$∵ A, B$ and $C$ are independent events, therefore $A^{'}, B$ and $C^{'}$ are independent events.

$\begin{aligned} \Rightarrow & 0.75 & = 1 - (1 - m) (1 - p) (1 - c) \\ \Rightarrow & 0.25 & = (1 - m) (1 - p) (1 - c) \end{aligned}$

Also, $P$ (passing exactly in two subjects) $= 0.4$

$\Rightarrow P (A \cap B \cap \bar{C} \cup A \cap \bar{B} \cap C \cup \bar{A} \cap B \cap C) = 0.4$

$\Rightarrow P (A \cap B \cap \bar{C}) + P (A \cap \bar{B} \cap C) + P (\bar{A} \cap B \cap C) = 0.4$

$\Rightarrow P (A) P (B) P (\bar{C}) + P (A) P (\bar{B}) P (C)$

$+ P (\bar{A}) P (B) P (C) = 0.4$

$\Rightarrow p m (1 - c) + p (1 - m) c + (1 - p) m c = 0.4$

$\Rightarrow p m - p m c + p c - p m c + m c - p m c = 0.4$

Again, $P$ (passing atleast in two subjects) $= 0.5$

$\Rightarrow P (A \cap B \cap \bar{C}) + P (A \cap \bar{B} \cap C)$

$+ P (\bar{A} \cap B \cap C) + P (A \cap B \cap C) = 0.5$

$\Rightarrow p m (1 - c) + p c (1 - m) + c m (1 - p) + p c m = 0.5$

$\Rightarrow p m - p c m + p c - p c m + c m - p c m + p c m = 0.5$

$\Rightarrow (p m + p c + m c) - 2 p c m = 0.5$

From Eq. (ii),

From Eq. (i),

$p m + p c + m c - 3 p c m = 0.4$

पिछला

अगला

Probability 3 Question 24

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

Probability 3 Question 24

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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