Let E ^ C denote the complement of an event E . Let E _ 1 , E _ 2...

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\left({E}_{1}\right) > 0

and

P\left({E}_{1} \cap {E}_{2} \cap {E}_{3}\right) = 0

then

P\left(\left({E}_{2}^{C} \cap {E}_{3}^{C}\right) / {E}_{1}\right)

is equal to

Options

A $P\left({E}_{2}^{C}\right) + P\left({E}_{3}\right)$
B $P\left({E}_{3}^{C}\right) - P\left({E}_{2}^{C}\right)$
C $P\left({E}_{3}\right) - P\left({E}_{2}^{C}\right)$
D $P\left({E}_{3}^{C}\right) - P\left({E}_{2}\right)$

Correct answer

D. $P\left({E}_{3}^{C}\right) - P\left({E}_{2}\right)$

Step-by-step solution

$P\left(\left({E}_{2}^{C} \cap {E}_{3}^{C}\right) / {E}_{1}\right) = \dfrac{P\left({E}_{1} \cap \left({E}_{2}^{C} \cap {E}_{3}^{C}\right)\right)}{P ( {E}_{1} )}$

$= \dfrac{P\left({E}_{1}\right) - \left\{P\left({E}_{1} \cap {E}_{2}\right) + P\left({E}_{1} \cap {E}_{3}\right) - P\left({E}_{1} \cap {E}_{2} \cap {E}_{3}\right)\right\}}{P\left({E}_{1}\right)}$

$= \dfrac{P\left({E}_{1}\right) - P\left({E}_{1} \cap {E}_{2}\right) - P\left({E}_{1} \cap {E}_{3}\right) - 0}{P\left({E}_{1}\right)}$

$= 1 - \dfrac{P\left({E}_{1} \cap {E}_{2}\right)}{P\left({E}_{1}\right)} - \dfrac{P\left({E}_{1} \cap {E}_{3}\right)}{P\left({E}_{1}\right)}$ = 1 - P $\left({E}_{2} / {E}_{1}\right) - P\left({E}_{3} / {E}_{1}\right)$

$= 1 - P\left({E}_{2}\right) - P\left({E}_{3}\right)$

$= P\left({E}_{3}^{C}\right) - P\left({E}_{2}\right)$ or $P\left({E}_{2}^{C}\right) - P\left({E}_{3}\right)$

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