Question

Statistic

Q4 (a) On a sunny day, a theme park had 1,000 visitors. According to the attendance record, 800 visitors took a ride on the roller coaster; 450 visitors took a ride on the merry-go-round. It is estimated that among those visitors who took a ride on the roller coaster, 40% of them also took a ride on the merry-go-round. A visitor on that day is selected at random.

i. What is the probability that this visitor rode on both rides?

ii. What is the probability that this visitor rode on no rides at all?

iii. If this visitor has taken a ride on the merry-go-round, what is the probability that he has not ridden on the roller coaster?

Q4(b) In a tutorial session, there are 11 Japanese students, 6 American students and 8 Australian students. Among these 25 students, a group of 5 students is selected randomly for the first presentation.

i. How many different groups can be formed?

ii. What is the probability that this group consists of only Japanese students?

iii. What is the probability that this group consists of exactly 2 Japanese students and 3 American students?

Q4(c) Three urns contain colored balls. Urn 1 contains 3 red, 4 white and 1 blue balls. Urn 2 contains 4 red, 3 white and 2 blue balls. Urn 3 contains 1 red, 2 white and 3 blue balls. One urn is chosen at random and a ball is drawn from it. If the ball is red, what is the probability that it came from Urn 3?

Answer 1

4.

(a)

Number of visitors = 1000

Number of visitors took a ride on the roller coaster = 800

It is given that 40% of the visitors who took a ride on the roller coaster also took a ride on the merry-go-round.

So, number of visitors took on roller coaster as well as merry-go-round = 800*0.40 = 320

So, number of visitors took a ride on the roller coaster only = 800-320 = 480

So, number of visitors took a ride on the merry-go-round only = 450-320 = 130

So, number of visitors who did not take any ride = 1000-(480+320+130) = 70

Suppose, R denotes the event that a visitor took a ride in roller coaster and M denotes the event that a visitor took a ride in merry-go-round.

(i)

Required probability is given by

(ii)

Required probability is given by

(iii)

Required probability is given by

(b)

(i)

Number of possible groups can be formed =

(ii)

Number of possible groups consisting only Japanese students =

Required probability

(iii)

Number of possible groups consisting 2 Japanese students and 3 American students =

Required probability

(c)

Suppose, U₁ denotes the event that a ball came from urn 1, U₂ denotes the event that a ball came from urn 2, U₃ denotes the event that a ball came from urn 3 and R denotes the event that a ball is red.

From the given data we have probabilities as follows.

Using Bayes' theorem required probability is given by