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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Probability

NDA 2021 · Mathematics · Probability

Consider the following relations for two events E and F : array ll...

Consider the following relations for two events EE and F:F : 1. P(E∩F)≥P(E)+P(F)−1 2. P(E∪F)=P(E)+P(F)+P(E∩F) 3. P(E∪F)≤P(E)+P(F)\begin{array}{ll} \text{ 1. } & P(E \cap F) \geq P(E)+P(F)-1 \\ \text{ 2. } & P(E \cup F)=P(E)+P(F)+P(E \cap F) \\ \text{ 3. } & P(E \cup F) \leq P(E)+P(F) \end{array}$$ Which of the above relations is/are correct?

Options

  1. A1 only
  2. B3 only
  3. C1 and 3 only
  4. D1, 2 and 3

Correct answer

C. 1 and 3 only

Step-by-step solution

Here⁡P(E∪F)=P(E)+P(F)−P(E∩F)\operatorname{Here} P(E \cup F)=P(E)+P(F)-P(E \cap F) We know 0≤P(E∪F)≤1⇒P(E)+P(F)−P(E0 \leq \mathrm{P}(\mathrm{E} \cup \mathrm{F}) \leq 1\Rightarrow P(E)+P(F)-P(E I F)≤1⇒1+P(EIF)≥P(E)+P(F)⇒P(E∩F)≥P(E)+P(F)−1F) \leq 1\Rightarrow 1+P(E I F) \geq\mathrm{P}(\mathrm{E})+\mathrm{P}(\mathrm{F})\Rightarrow \mathrm{P}(\mathrm{E} \cap \mathrm{F}) \geq \mathrm{P}(\mathrm{E})+\mathrm{P}(\mathrm{F})-1 Again, P(E∪F)=P(E)+P(F)−P(EIF)⇒P(E∪F)+P(E∩F)=P(E)+P(F)⇒P(E)+P(F)=P(E∪F)+P(E∩F)⇒P(E)+P(F)≥P(E∪F)P(E \cup F)=P(E)+P(F)-P(E I F)\Rightarrow \mathrm{P}(\mathrm{E} \cup \mathrm{F})+\mathrm{P}(\mathrm{E} \cap \mathrm{F})=\mathrm{P}(\mathrm{E})+\mathrm{P}(\mathrm{F})\Rightarrow \mathrm{P}(\mathrm{E})+\mathrm{P}(\mathrm{F})=\mathrm{P}(\mathrm{E} \cup \mathrm{F})+\mathrm{P}(\mathrm{E} \cap \mathrm{F})\Rightarrow \mathrm{P}(\mathrm{E})+\mathrm{P}(\mathrm{F}) \geq \mathrm{P}(\mathrm{E} \cup \mathrm{F}) Hence Relations (1) and (3) are correct.
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