यदि  और है, तो निम्नलिखित में से कौन-सा सही नहीं है?

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NDA (Held On: 6 Sep 2020) Maths Previous Year paper
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  1. P(A ∩ B) = P(A) P(B)
  2. P(A ∪ B) > P(A) + P(B)
  3. P(ना तो A और ना ही B ) = P(ना A) P(ना B)

Answer (Detailed Solution Below)

Option 3 : P(A ∪ B) > P(A) + P(B)
Free
NDA 01/2025: English Subject Test
30 Qs. 120 Marks 30 Mins

Detailed Solution

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संकल्पना:

एक घटना का पूरक:

एक घटना का पूरक प्रतिदर्श समष्‍टि में परिणामों का उपसमुच्चय होता है जो घटना में नहीं होते हैं। 

एक घटना के पूरक की प्रायिकता घटना के प्रायिकता से एक कम है। 

P (A̅) = 1 - P (A) ⇒ P (A) = 1 - P (A̅)

सूत्र: P (A ∪ B) =  P (A) + P (B) - P (A ∩ B)

 

गणना:

दिया गया है:

इसलिए, P(A) = 1 - P (A̅) = 

चूँकि हम जानते हैं, P (A ∪ B) =  P (A) + P (B) - P (A ∩ B)

⇒ P (A ∪ B) =  1 - P (A̅) + P (B) - P (A ∩ B)

इसलिए विकल्प 1 सत्य है। 

 

P(A ∩ B) = P(A) P(B) = 

इसलिए, विकल्प 2 भी सत्य है। 

 

P(A) + P(B) =  >  P(A ∪ B) 

अतः विकल्प 3 सत्य नहीं है। 

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