Question

Suppose a study of speeding violations and drivers who use cell phones produced the following fictional data: Table 3.2.

	Speeding violation in the last year	No speeding violation in the last year	Total
Uses cell phone while driving	25	280	305
Does not use cell phone while driving	45	405	450
Total	70	685	755

The total number of people in the sample is 755. The row totals are 305 and 450. The column totals are 70 and

685. Notice that 305 + 450 = 755 and 70 + 685 = 755.

Calculate the following probabilities using the table.

a. Find P(Driver is a cell phone user).

b. Find P(driver had no violation in the last year).

c. Find P(Driver had no violation in the last year AND was a cell phone user).

d. Find P(Driver is a cell phone user OR driver had no violation in the last year).

e. Find P(Driver is a cell phone user GIVEN driver had a violation in the last year).

f. Find P(Driver had no violation last year GIVEN driver was not a cell phone user)

Short answer

Part a: P(Driver is a cell phone user) =$\frac{305}{755}$

Part b: P(driver had no violation in the last year) =$\frac{685}{755}$

Part c: P(Driver had no violation in the last year AND was a cell phone user) =$\frac{280}{755}$

Part d: P(Driver is a cell phone user OR driver had no violation in the last year) =$\frac{710}{755}$

Part e: P(Driver is a cell phone user GIVEN driver had a violation in the last year) =$\frac{25}{70}$

Part f: P(Driver had no violation last year GIVEN driver was not a cell phone user) =$\frac{405}{450}$

Step-by-step solution

Given information (Part a)

number of cell phone users = 305, Total number of users in study = 755.

Probability of drivers as cell phone users

Part a:$\frac{numberofcellphoneusers}{totalnumberinstudy}=\frac{305}{755}$

Given information (Part b)

number that had no violation = 685, Total number of users in study = 755.

P(driver had no violation in the last year)  

Part b:$\frac{numberthathadnoviolation}{totalnumberinstudy}=\frac{685}{755}$

Given information (Part c)

the number had no violation in the last year AND was a cell phone user = 280, total number of study = 755

P(Driver had no violation in the last year AND was a cell phone user) 

Part c:$\frac{numberofdriverhadnoviolationinlastyearandcellphoneuser}{totalnumberofstudy}=\frac{280}{755}$

Given information (Part d)

Probability of driver is a cell phone user OR driver had no violations in the last year = Probability of driver as cell phone user + Probability of driver had no violation last year - Probability of driver had no violation last year AND was a cell phone user

Probability of driver is a cell phone user OR driver had no violations in the last year 

Part d:$\frac{305}{755}+\frac{685}{755}-\frac{280}{755}=\frac{710}{755}$

Given information (Part e)

number of drivers had violations in the last year while using cell phones = 25. Total drivers had violations in the last year = 70.

Step 2: P (Driver is a cell phone user GIVEN driver had a violation in the last year)

part e:$\frac{25}{70}$

Given information (Part f)

P(Driver had no violation last year GIVEN driver was not a cell phone user), number of the driver had no violation last year but not a cell phone user = 405 Total number of the driver was not a cell phone user = 450

P(Driver had no violation last year GIVEN driver was not a cell phone user) 

part f:$\frac{405}{450}$