Mathematics - Probability Question with Solution | TestHub

MathematicsProbabilityConditional probabilityHard2 minPYQ_2020

MathematicsHardsingle choice

Let $E^{C}$ denote the complement of an event $E$ . Let $E_{1}, E_{2}$ and $E_{3}$ be any pairwise independent events with $P (E_{1}) > 0$ and $P (E_{1} \cap E_{2} \cap E_{3}) = 0$ then $P ((E_{2}^{C} \cap E_{3}^{C}) / E_{1})$ is equal to

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Answer:

Solution:

$P ((E_{2}^{C} \cap E_{3}^{C}) / E_{1}) = \frac{P (E_{1} \cap (E_{2}^{C} \cap E_{3}^{C}))}{P (E_{1})}$

$= \frac{P (E_{1}) - \{P (E_{1} \cap E_{2}) + P (E_{1} \cap E_{3}) - P (E_{1} \cap E_{2} \cap E_{3})\}}{P (E_{1})}$

$= \frac{P (E_{1}) - P (E_{1} \cap E_{2}) - P (E_{1} \cap E_{3}) - 0}{P (E_{1})}$

$= 1 - \frac{P (E_{1} \cap E_{2})}{P (E_{1})} - \frac{P (E_{1} \cap E_{3})}{P (E_{1})} = 1 - P (E_{2} / E_{1}) - P (E_{3} / E_{1})$

$= 1 - P (E_{2}) - P (E_{3})$

$= P (E_{3}^{C}) - P (E_{2})$ or $P (E_{2}^{C}) - P (E_{3})$

Stream:JEESubject:MathematicsTopic:ProbabilitySubtopic:Conditional probability

⏱ 2mℹ️ Source: PYQ_2020

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