Question

For two events, A and B, P(A)= .4, P(B)= .2 , and $\mathbf{\text{P(A}}\mathbf{\cap }\mathbf{\text{B) = .1}}$:

a. Find P (A/B).

b. Find P(B/A).

c. Are A and B independent events?

Short answer

Answer

1. 0.50
2. 0.25
3. No

Step-by-step solution

Step-by-Step SolutionStep 1: Introduction

The likelihood of an event occurring if another event has already occurred is conditional probability. The conditional probability formula:

$\mathbf{\text{P (A/B) =}}\frac{\mathbf{\text{P (A}}\mathbf{\cap }\mathbf{\text{B)}}}{\mathbf{\text{P (B)}}}$

Find the probability

$\begin{array}{c}\mathbf{\text{P (A/B) =}}\frac{\mathbf{\text{P (A}}\mathbf{\cap }\mathbf{\text{B)}}}{\mathbf{\text{P (B)}}}\\ \mathbf{\text{=}}\frac{\mathbf{\text{.1}}}{\mathbf{\text{.2}}}\\ \mathbf{\text{=.50}}\end{array}$

Hence, the required probability is 0.50.

Find the probability

$\begin{array}{c}\mathbf{\text{P (B/A) =}}\frac{\mathbf{\text{P (B}}\mathbf{\cap }\mathbf{\text{A)}}}{\mathbf{\text{P (A)}}}\\ \mathbf{\text{=}}\frac{\mathbf{\text{.1}}}{\mathbf{\text{.4}}}\\ \mathbf{\text{=.25}}\end{array}$

Hence, the required probability is 0.25.

Identify that A and B are independent events

For independent event $\mathbf{\text{P (A}}\mathbf{\cap }\mathbf{\text{B) = P(A)}}\mathbf{\times }\mathbf{\text{P(B)}}$

Here$\mathbf{\text{P(A)}}\mathbf{\times }\mathbf{\text{P(B) = .4}}\mathbf{\times }\mathbf{\text{.2= 0.08}}$, which means $\mathbf{\text{P (A}}\mathbf{\cap }\mathbf{\text{B) = 0.1}}\mathbf{\ne }\mathbf{\text{0.08}}$.

No, A and B are not independent events.