Chapter 5, Problem T5.3SPT

To determine

To check: Whether the two events “Owner has a Chevy” and “Owner’s truck has four wheel drive” are independent and mutually exclusive.

Answer to Problem T5.3SPT

The two events “Owner has a Chevy” and “Owner’s truck has four wheel drive” are independent but not mutually exclusive. Option c is the correct choice.

Explanation of Solution

Given information:

Total number of truck owners $=125$

	Four wheel drive
Brand of truck		Yes	No
	Ford	28	17
	Chevy	32	18
	Dodge	20	10

Formula used:

Condition for independence

P (Owner has a Chevy $\cap$ Owner’s truck has four − wheel drive)

= P (Owner has a Chevy) $\times$ P (Owner’s truck has four − wheel drive)

  $\begin{array}{l}P\left(\text{Owner has a Chevy}\right)=\frac{\text{Number of trucks brand Chevy}}{\text{Total number of truck owners}}\\ P\left(\text{Owner}’\text{s truck has four wheel drive}\right)=\frac{\text{Number of four wheel drive}}{\text{Total number of truck owners}}\\ \text{P }(\text{Owner has a Chevy }\cap \text{ Owner}’\text{s truck has four wheel drive})=\frac{\text{Number of four wheel drive with brand Chevy}}{\text{Total number of truck owners}}\end{array}$

Calculation:

Checking independence

P (Owner has a Chevy) $=\frac{32+18}{125}=\frac{50}{125}$

P (Owner’s truck has four − wheel drive) $=\frac{28+32+20}{125}=\frac{80}{125}$

P (Owner has a Chevy $\cap$ Owner’s truck has four − wheel drive) $=\frac{32}{125}$

Checking condition of independence - $\frac{32}{125}=\frac{50}{125}\times \frac{80}{125}$

For the events to be mutually exclusive, both should not overlap. This means that both events should not occur at the same time. In this case, the events “Owner has a Chevy” and “Owner’s truck has four − wheel drive” are overlapping.