Question

Cell phone handoff behaviour. A “handoff” is a term used in wireless communications to describe the process of a cell phone moving from the coverage area of one base station to that of another. Each base station has multiple channels (called color codes) that allow it to communicate with the cell phone. The Journal of Engineering, Computing and Architecture (Vol. 3., 2009) published a cell phone handoff behavior study. During a sample driving trip that involved crossing from one base station to another, the different color codes accessed by the cell phone were monitored and recorded. The table below shows the number of times each color code was accessed for two identical driving trips, each using a different cell phone model. (Note: The table is similar to the one published in the article.) Suppose you randomly select one point during the combined driving trips.

Color code
	0	5	b	c	Total
Model 1	20	35	40	0	85
Model 2	15	50	6	4	75
Total	35	85	46	4	160

a. What is the probability that the cell phone was using color code 5?

b. What is the probability that the cell phone was using color code 5 or color code 0?

c. What is the probability that the cell phone used was Model 2 and the color code was 0?

Short answer

1. 0.53
2. 0.75
3. 0.94

Step-by-step solution

Introduction

The probability of an occurrence refers to the possibility that the event will occur. The formula represents as:

$\mathbf{P}\mathbf{\left(}\mathbf{E}\mathbf{\right)}\mathbf{\text{}}\mathbf{=}\frac{\mathbf{Favourableoutcome}}{\mathbf{Totaloutcome}}$

Find the probability of color code 5

$$\begin{gathered} \mathbf{P}\mathbf{(}\mathbf{Color}\mathbf{}\mathbf{code}\mathbf{}\mathbf{5}\mathbf{)}\mathbf{=}\frac{\mathbf{Favorable}\mathbf{}\mathbf{outcome}}{\mathbf{Total}\mathbf{}\mathbf{outcome}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{85}}{\mathbf{160}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{53} \end{gathered}$$

Hence, the probability of color code 5 is 0.53.

Find the probability of color code 5 or 0

$\mathbf{P}\mathbf{(}\mathbf{Color}\mathbf{}\mathbf{code}\mathbf{}\mathbf{5}\mathbf{}\mathbf{or}\mathbf{}\mathbf{0}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{(}\mathbf{color}\mathbf{}\mathbf{code}\mathbf{}\mathbf{5}\mathbf{)}\mathbf{+}\mathbf{P}\mathbf{(}\mathbf{color}\mathbf{}\mathbf{code}\mathbf{}\mathbf{0}\mathbf{)}\mathbf{-}\mathbf{P}\mathbf{(}\mathbf{color}\mathbf{}\mathbf{code}\mathbf{}\mathbf{5}\mathbf{}\mathbf{and}\mathbf{}\mathbf{0}\mathbf{)}$

$$\begin{gathered} \mathbf{P}\mathbf{(}\mathbf{color}\mathbf{}\mathbf{code}\mathbf{0}\mathbf{)}\mathbf{=}\frac{\mathbf{Favorable}\mathbf{}\mathbf{outcome}}{\mathbf{Total}\mathbf{}\mathbf{out}\mathbf{}\mathbf{come}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{35}}{\mathbf{160}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{22} \end{gathered}$$

$\mathbf{P}\mathbf{(}\mathbf{color}\mathbf{}\mathbf{code}\mathbf{5}\mathbf{}\mathbf{and}\mathbf{}\mathbf{0}\mathbf{)}\mathbf{=}\mathbf{0}$

$\mathbf{P}\mathbf{(}\mathbf{Color}\mathbf{}\mathbf{code}\mathbf{}\mathbf{5}\mathbf{}\mathbf{or}\mathbf{}\mathbf{0}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{53}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{22}\mathbf{-}\mathbf{0}$

Hence, the probability of color code 5 or 0 is 0.75.

Find the probability of color code Model 2 and color code 0

$$\begin{gathered} \mathbf{P}\mathbf{(}\mathbf{Mode}\mathbf{}\mathbf{l}\mathbf{2}\mathbf{}\mathbf{and}\mathbf{}\mathbf{colorcode}\mathbf{}\mathbf{0}\mathbf{)}\mathbf{=}\frac{\mathbf{15}}{\mathbf{160}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{94} \end{gathered}$$

Hence, the probability of Model 2 and color code 0 is 0.94.