3 Paying monthly bills online. Do most people pay their monthly bills online? ABA Bank Marketing and Sales (July-August 2015) reported that 37% of U.S. customers pay their bills online using a personal computer. The table lists the different methods by which customers pay their monthly bills and associated probabilities. Consider the following two events:

A = {Customer does not pay monthly bills online.}

B = {Customer does not pay monthly bills with a credit or debit card.}

Payment Method	Probability
Online via personal computer	.37
Write check	.22
Checking account withdrawal	.10
Debit card	.09
Credit card	.07
Cash	.08
Mobile bill account	.04
Online via tablet or e-reader	.03

a. Find $\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}$.

b. Find$\mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)}$.

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

d. Use the additive law of probability to find $\mathbf{P}\mathbf{\text{}}\mathbf{(}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{)}$ .

Step by step solution

01

Introduction

Probability is a really basic idea. It denotes the likelihood of an event occurring or the possibility of an event occurring. Probability is calculated as follows:

$\mathbf{P}\mathbf{\text{}}\mathbf{(}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{-}\mathbf{P}\mathbf{\text{}}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{)}$

02

Find P (A)

$$\begin{gathered} \mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{22}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{10}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{09}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{07}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{08} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{56} \end{gathered}$$

Hence, the required probability is0.56.

03

Find P (B)

$$\begin{gathered} \mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{37}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{22}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{10}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{08} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{77} \end{gathered}$$

Hence, the required probability is0.77.

04

Find the required probability

$$\begin{gathered} \mathbf{P}\mathbf{\text{}}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{22}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{10}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{08} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{40} \end{gathered}$$

Hence, the required probability is0.40.

05

Find the required probability

$$\begin{gathered} \mathbf{P}\mathbf{\text{}}\mathbf{(}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{-}\mathbf{P}\mathbf{\text{}}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{56}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{77}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{40} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{93} \end{gathered}$$

Hence, the required probability is 0.93.

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