Question

Let E^C denote the complement of an event E. Let E₁, E₂ and E₃ be any pairwise independent events with P(E₁) > 0 and P(E₁ ∩ E₂ ∩ E₃) = 0. Then `"P"(("E"_2^"C"  ∩ "E"_3^"C")/"E"_1)` is equal to ______.

Options

• `"P"("E"_2^"C") + "P"("E"_3)`

• `"P"("E"_3^"C") - "P"("E"_2^"C")`

• `"P"("E"_3) - "P"("E"_2^"C")`

• `"P"("E"_3^"C") - "P"("E"_2)`

Answer 1

Let E^C denote the complement of an event E. Let E₁, E₂ and E₃ be any pairwise independent events with P(E₁) > 0 and P(E₁ ∩ E₂ ∩ E₃) = 0. Then `"P"(("E"_2^"C"  ∩ "E"_3^"C")/"E"_1)` is equal to `underlinebb("P"("E"_3^"C") - "P"("E"_2))`.

Explanation:

Given, E₁, E₂, E₃ are pairwise independent events

So, P(E₁ ∩ E₂) = P(E₁).P(E₂);

P(E₂ ∩ E₃) = P(E₂).P(E₃);

P(E₃ ∩ E₁) = P(E₃).P(E₁);

and P(E₁ ∩ E₂ ∩ E₃) = 0

`"P"(("E"_2^"C" ∩ "E"_3^"C")/"E"_1) = ("P"["E"_1 ∩ ("E"_2^"C" ∩ "E"_3^"C")])/("P"("E"_1))`

= `("P"("E"_1) - "P"["E"_1 ∩ ("E"_2 ∪ "E"_3)])/("P"("E"_1))`  ...[∵ P(A ∩ B^C) = P(A) – P(A) – P(A ∩ B)]

= `("P"("E"_1) - "P"[("E"_1 ∩ "E"_2) ∪ ("E"_1 ∩ "E"_3)])/("P"("E"_1))`

= `("P"("E"_1) - ["P"("E"_1 ∩ "E"_2) + "P"("E"_1 ∩ "E"_3) - "P"("E"_1 ∩ "E"_2 ∩ "E"_3)])/("P"("E"_1))`

= `("P"("E"_1) - "P"("E"_1 ∩ "E"_2) - "P"("E"_1 ∩ "E"_3) + 0)/("P"("E"_1))`

= 1 – P(E₂) – P(E₃)  ...[∵ P(A ∩ B) = P(A).P(B)]

= `"P"("E"_2^"C") - "P"("E"_3)` or `"P"("E"_3^"C") - "P"("E"_2)`