# For two events A and B, P(A)=(4/5), P(B|A)=(1/2) ,also know that A and B are NOT...

## Question:

For two events {eq}A {/eq} and {eq}B, P(A)=(4/5), P(B|A)=(1/2) {/eq},also know that {eq}A {/eq} and {eq}B {/eq} are NOT independent.

For each statement select either "True", "False", or "Can't Tell", and give a reason for your answer.

a. {eq}A {/eq} and {eq}B {/eq} are mutually exclusive.

b. {eq}A {/eq} and are independent

c. {eq}P(A \cap B) > P(A) \times P(B) {/eq}

d. {eq}P(B) \leq P(A) {/eq}

## Mutually exclusive and Independent events

Events are said to be mutually exclusive when the joint probability of events is the product of the marginal probability and events are said to be independent when the intersection or joint probability of two events is equal to 0.

## Answer and Explanation:

Given information:

{eq}\begin{align*} P\left( A \right) &= \dfrac{4}{5}\\ P\left( {B\left| A \right.} \right) &= \dfrac{1}{2}\\ P\left( {A \cap B} \right) &\ne P\left( A \right) \times P\left( B \right) \end{align*} {/eq}

a

A and B is mutually exclusive is tested as follows:

{eq}\begin{align*} P\left( {A \cap B} \right) &= P\left( {B\left| A \right.} \right) \times P\left( A \right)\\ &= \dfrac{1}{2} \times \dfrac{4}{5} \Rightarrow \dfrac{2}{5} \end{align*} {/eq}

As, probability of intersection is not equal to zero it is concluded that statement A and B are not mutually exclusive events is False.

B

False. because of the following reason:

{eq}P\left( {A \cap B} \right) \ne P\left( A \right) \times P\left( B \right) {/eq}

C

Probability of B is calculated as follows:

{eq}\begin{align*} P\left( B \right) &= 1 - P\left( A \right) + P\left( {A \cap B} \right)\\ &= 1 - 0.8 + 0.4 \Rightarrow 0.6 \end{align*} {/eq}

Hence,

{eq}\begin{align*} P\left( {A \cap B} \right) &= 0.4\\ P\left( A \right) \times P\left( B \right) &= 0.8 \times 0.6 \Rightarrow 0.48\\ P\left( {A \cap B} \right) &< P\left( A \right) \times P\left( B \right) \end{align*} {/eq}

Hence, the above statement is False.

D

As,

{eq}\begin{align*} P\left( B \right) &= 0.6\\ P\left( A \right) &= 0.8\\ P\left( B \right) &< P\left( A \right) \end{align*} {/eq}

Hence, the above statement is true.