Answer to Question #191184 in Statistics and Probability for Dani

Question

1)Given P (A) = 0.30, P (B) = 0.78, P (A∩ B) = 0.16; evaluate

A) A complement intersection B complement

B) A complement union B complement

C) A intersection B complement

 

Answer 1

A) Let's find

$p(\overline A ) = 1 - p(A) = 1 - 0.3 = 0.7$

$p(\overline B ) = 1 - p(B) = 1 - 0.78 = 0.22$

$p(\overline {A \cap B} ) = p\left( {\overline A \cup \overline B } \right) = 1 - 0.16 = 0.84$

Since $p\left( {\overline A \cup \overline B } \right) = p(\overline A ) + p(\overline B ) - p(\overline A \cap \overline B )$ then

$p(\overline A \cap \overline B ) = p(\overline A ) + p(\overline B ) - p\left( {\overline A \cup \overline B } \right) = 0.7 + 0.22 - 0.84 = 0.08$

Answer: $p(\overline A \cap \overline B ) = 0.08$

B) $p\left( {\overline A \cup \overline B } \right) = p(\overline {A \cap B} ) = 1 - 0.16 = 0.84$

Answer: $p\left( {\overline A \cup \overline B } \right) = 0.84$

C) $p(A \cap \overline B ) = p(A\backslash B) = p(A) - p(A \cap B) = 0.3 - 0.16 = 0.14$

Answer: $p(A \cap \overline B ) = 0.14$