एक जहाज में तीन इंजन E_1, E_2 और E_3 लगे हैं। | vedclass

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(A) दी गई प्रायिकताएँ: $P(X_1) = \frac{1}{2}, P(X_2) = \frac{1}{4}, P(X_3) = \frac{1}{4}$।मान लीजिए $X_1^c, X_2^c, X_3^c$ वे घटनाएँ हैं कि इंजन कार्य

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$(B, D)$

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(A) दी गई प्रायिकताएँ: $P(X_1) = \frac{1}{2}, P(X_2) = \frac{1}{4}, P(X_3) = \frac{1}{4}$।मान लीजिए $X_1^c, X_2^c, X_3^c$ वे घटनाएँ हैं कि इंजन कार्य नहीं कर रहे हैं,इसलिए $P(X_1^c) = \frac{1}{2}, P(X_2^c) = \frac{3}{4}, P(X_3^c) = \frac{3}{4}$।जहाज चालू $(X)$ है यदि कम से कम दो इंजन कार्य करते हैं:$P(X) = P(X_1 X_2 X_3^c) + P(X_1 X_2^c X_3) + P(X_1^c X_2 X_3) + P(X_1 X_2 X_3)$$P(X) = (\frac{1}{2} \cdot \frac{1}{4} \cdot \frac{3}{4}) + (\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{1}{4}) + (\frac{1}{2} \cdot \frac{1}{4} \cdot \frac{1}{4}) + (\frac{1}{2} \cdot \frac{1}{4} \cdot \frac{1}{4}) = \frac{3}{32} + \frac{3}{32} + \frac{1}{32} + \frac{1}{32} = \frac{8}{32} = \frac{1}{4}$।$(A) P(X_1^c \mid X) = \frac{P(X_1^c \cap X)}{P(X)} = \frac{P(X_1^c X_2 X_3)}{P(X)} = \frac{\frac{1}{2} \cdot \frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{4}} = \frac{1}{8} 
eq \frac{3}{16}$। (असत्य)$(B) P(	ext{ठीक दो} \mid X) = \frac{P(X_1 X_2 X_3^c) + P(X_1 X_2^c X_3) + P(X_1^c X_2 X_3)}{P(X)} = \frac{\frac{3}{32} + \frac{3}{32} + \frac{1}{32}}{\frac{1}{4}} = \frac{7/32}{1/4} = \frac{7}{8}$। (सत्य)$(C) P(X \mid X_2) = \frac{P(X \cap X_2)}{P(X_2)} = \frac{P(X_1 X_2 X_3^c) + P(X_1^c X_2 X_3) + P(X_1 X_2 X_3)}{P(X_2)} = \frac{\frac{3}{32} + \frac{1}{32} + \frac{1}{32}}{\frac{1}{4}} = \frac{5/32}{1/4} = \frac{5}{8} 
eq \frac{5}{16}$। (असत्य)$(D) P(X \mid X_1) = \frac{P(X \cap X_1)}{P(X_1)} = \frac{P(X_1 X_2 X_3^c) + P(X_1 X_2^c X_3) + P(X_1 X_2 X_3)}{P(X_1)} = \frac{\frac{3}{32} + \frac{3}{32} + \frac{1}{32}}{\frac{1}{2}} = \frac{7/32}{1/2} = \frac{7}{16}$। (सत्य)

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