Question

For two independent events, A and B, P (A) = .4 and P(B) = .2 :

a. Find $\mathbf{\text{P (A∩B)}}$

b. Find P (A/B)

c. Find $\mathbf{\text{P (A}}\mathbf{\cup }\mathbf{\text{B)}}$

Short answer

Answer

1. 0.08
2. 0.4
3. 0.52

Step-by-step solution

Step-by-Step SolutionStep 1: Introduction

There are two occurrences, A and B. If the outcome of the A event has no impact on the outcome of the B, the two occurrences are said to be independent.

$\mathbf{\text{P}}\mathbf{(}\mathbf{\text{A}}\mathbf{\cap }\mathbf{\text{B}}\mathbf{)}\mathbf{\text{= P}}\mathbf{\left(}\mathbf{\text{A}}\mathbf{\right)}\mathbf{\times }\mathbf{\text{P}}\mathbf{\left(}\mathbf{\text{B}}\mathbf{\right)}$

Find the required probability

Since A and B are independent events.

Therefore,

$\begin{array}{c}\mathbf{\text{P (A}}\mathbf{\cap }\mathbf{\text{B) = P(A)}}\mathbf{\times }\mathbf{\text{P(B)}}\\ \mathbf{\text{= .4}}\mathbf{\times }\mathbf{\text{.2}}\\ \mathbf{\text{= .08}}\end{array}$

Hence, the required probability is 0.08.

Find the required probability

$\begin{array}{c}\mathbf{\text{P (A/B) =}}\frac{\mathbf{\text{P (A}}\mathbf{\cap }\mathbf{\text{B)}}}{\mathbf{\text{P (B)}}}\\ \mathbf{\text{=}}\frac{\mathbf{\text{.08}}}{\mathbf{\text{.2}}}\\ \mathbf{\text{=.4}}\end{array}$

Hence, the required probability is 0.4.

Find the required probability

$\begin{array}{c}\mathbf{\text{P (A}}\mathbf{\cup }\mathbf{\text{B) = P(A) + P (B)}}\mathbf{-}\mathbf{\text{P(A}}\mathbf{\cap }\mathbf{\text{B)}}\\ \mathbf{\text{= 0.4 + 0.2}}\mathbf{-}\mathbf{\text{0.08}}\\ \mathbf{\text{= 0.52}}\end{array}$

Hence, the required probability is 0.52.