Question

A bicycle company makes two mountain bike models that each come in three colors. Use the following​ table, which shows the production volumes for one​ week, to answer parts a through d.

	Color
Model	Blue	Brown	White
XK​-50	296	85	204
HD​-99	46	210	132

a. Based on the relative frequency assessment​ method, what is the probability that a manufactured item is brown​? ​P(brown​)=

​(Round to four decimal places as​ needed.)

b. What is the probability that the product manufactured is an XK-50​? ​P(XK-50​)=

​(Round to four decimal places as​ needed.)

c. What is the probability that a product manufactured is an XK-50 and brown​? ​P(XK-50 and brown​)=

​(Round to four decimal places as​ needed.)

d. Suppose a product was chosen at random. Consider the following​ events: the event that model HD-99 was chosen and the event that a blue product was chosen. Are these two events mutually​ exclusive? Explain.

A.The two events are mutually exclusive because the probability of the events occurring together is not 0.

B.The two events are not mutually exclusive because the probability of the events occurring together is not 0.

C.The two events are not mutually exclusive because the probability of the events occurring together is 0.

D.The two events are mutually exclusive because the probability of the events occurring together is 0.

A marketing company has randomly surveyed 213 men who watch professional sports. The men were separated according to their educational level​ (college degree or​ not) and whether they preferred the NBA or the National Football League​ (NFL). The results of the survey are shown to the right. Complete parts a through d.

Sports Preference College Degree No College Degree NBA 41 54 NFL 19 99

a. What is the probability that a randomly selected survey participant prefers the​ NFL?

​(Round to four decimal places as​ needed.)

b. What is the probability that a randomly selected survey participant has a college degree and prefers the​ NBA?

​(Round to four decimal places as​ needed.)

c. Suppose a survey participant is randomly selected and you are told that he has a college degree. What is the probability that this man prefers the​ NFL?

​(Round to four decimal places as​ needed.)

d. Is a survey​ participant's preference for the NBA independent of having a college​ degree?

yes. The two events are independent.

No. The two events are not independent.

Answer 1

Solution : Given that,

A bicycle company makes two mountain bike models that each come in three colors. Data is as follows

	Color
Model	Blue	Brown	White
XK​-50	296	85	204
HD​-99	46	210	132

Here the total number of items = 973

a. Based on the relative frequency assessment​ method, what is the probability that a manufactured item is brown​? ​P(brown​)= n( brown items ) / total number of items

= (85+210) / 973

= 0.303186

​the probability that a manufactured item is brown is 0.3032

b. What is the probability that the product manufactured is an XK-50​?

​P(XK-50​)=n( XK-50 items ) / total number of items

​ = (296+85+204) /973

= 0.60123

the probability that the product manufactured is an XK-50​ is 0.6012

c. What is the probability that a product manufactured is an XK-50 and brown​?

​P(XK-50 and brown​) = n( brown items and XK-50 ) / total number of items

= 85/973

= 0.08735

the probability that a product manufactured is an XK-50 and brown​ is 0.0874

d. Suppose a product was chosen at random. Consider the following​ events: the event that model HD-99 was chosen and the event that a blue product was chosen. Are these two events mutually​ exclusive? Explain.

We know that if the two events are mutually exclusive then,

p(A $\cap$ B ) =0

There must not be common anything

Here the item can be HD-99 and blue in color at a time and probability of this is not zero.

B.The two events are not mutually exclusive because the probability of the events occurring together is not 0.

Thus option B) is correct

2.

A marketing company has randomly surveyed 213 men who watch professional sports. The men were separated according to their educational level​ (college degree or​ not) and whether they preferred the NBA or the National Football League​ (NFL). The results of the survey are shown to the right. Complete parts a through d.

Sports Preference College Degree No College Degree NBA 41 54 NFL 19 99

a. What is the probability that a randomly selected survey participant prefers the​ NFL?

​P(NFL) = n(NFL participants ) / n(total perticipants)

= (19+99) / 213

= 0.55399

the probability that a randomly selected survey participant prefers the​ NFL is 0.5540

b. What is the probability that a randomly selected survey participant has a college degree and prefers the​ NBA?

​P(NBA$\cap$ college degree ) = n(NBA and college degree participants ) / n(total perticipants)

= 41/213

= 0.192488

the probability that a randomly selected survey participant has a college degree and prefers the​ NBA is 0.1923

c. Suppose a survey participant is randomly selected and you are told that he has a college degree. What is the probability that this man prefers the​ NFL?

​P(NFL $\cap$ college degree) = n(NFL and college degree participants ) / n(total perticipants)

= 19 /213

= 0.08920

the probability that this man prefers the​ NFL is 0.0892

d. Is a survey​ participant's preference for the NBA independent of having a college​ degree?

No. The two events are not independent.

Since they can occure simultaneusly they are not mutually exclusive. They have common participants. Thus a survey​ participant's preference for the NBA independent of having a college​ degree is not independent of each other.