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 $P(\frac{{E_2^C \cap E_3^C}}{E_1})$ is equal to

Question

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 $P(\frac{{E_2^C \cap E_3^C}}{E_1})$ is equal to

Aicharya · Accepted Answer

P E 2 C &#8745; E 3 C / E 1 = P E 1 &#8745; E 2 C &#8745; E 3 C P ( E 1 ) = P E 1 - P E 1 &#8745; E 2 + P E 1 &#8745; E 3 - P E 1 &#8745; E 2 &#8745; E 3 P E 1 = P E 1 - P E 1 &#8745; E 2 - P E 1 &#8745; E 3 - 0 P E 1 = 1 - P E 1 &#8745; E 2 P E 1 - P E 1 &#8745; 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

Let $E^C$ denote the complement of an event $E$. Let $E_1$, $E_2$ and $E_3$ be any pairwise independent…

Solution