Probability and Statistics - Online Test

Q1.
Direction: Study the following information carefully and answer the questions that follow:

A committee of 10 persons is to be formed from 7 men and 6 women.
In how many ways this can be done if at least 5 men to have to be included in a committee.
Answer : Option A
Explaination / Solution:

Number of Ways when if at least 5 men include in committee = 7C5*6C5+7C6*6C4+7C7*6C3 
 = 21*6+7*15+1*20
= 251

Q2.
Direction: Study the following information carefully and answer the questions that follow:

A committee of 10 persons is to be formed from 7 men and 6 women.
In how many ways of these committee the women are in majority
Answer : Option B
Explaination / Solution:

Number of ways when women are majority in committee = 6C6*7C4 = 1*35 = 35

Q3.
Direction: Study the following information carefully and answer the questions that follow:

A committee of 10 persons is to be formed from 7 men and 6 women.
In how many of these committee the men are in majority
Answer : Option D
Explaination / Solution:

Number of ways when men are majority in committee = 7C6*6C4+7C7*6C3 
7*15+1*20 = 125

Q4.
Direction: Study the following information carefully and answer the questions that follow:

A committee of 10 persons is to be formed from 7 men and 6 women.
In how many ways this can be done if 6 men and 4 women be included in a committee
Answer : Option A
Explaination / Solution:

If 6 men and 4 women include in committee = 7C6*6C4 --> 7*15 --> 105

Q5.
Direction: Study the following information carefully and answer the questions that follow:

A committee of 10 persons is to be formed from 7 men and 6 women.
In how many ways this can be done if 5 men and 5 women be included in a committee
Answer : Option D
Explaination / Solution:

If 5 men and 5 women include in committee = 7C5*6C5 --> 21*6 = 126

Q6. In a basket there are 7 apples and 8 oranges. 4 fruits are picked at random. What is the probability that two fruits are apples and 2 are oranges?
Answer : Option E
Explaination / Solution:

Total number of fruits n(s) =7 + 8=15 
Probability  = (7C2*8C2)/15C4
= (21*28)/1365 = 28/65

Q7. In how many ways the letter of the word IMMEDIATELY can be arranged so that vowels always come together ?
Answer : Option C
Explaination / Solution:

There are 11 letters .out of which 5 are vowels and 6 are consonants. 
So taking vowels together as a single letter , we have 7 letters 
So no of arrangements =( 7! * 5!)/ (2!)3 [there are 2 i,m e]

Q8. In how many ways can the word ENGINEER be arranged so that ‘G’ and ‘R’ are never together?
Answer : Option C
Explaination / Solution:

Number of ways of rearranging the word ENGINEER  = (8!)/(3!*2!) = 3360 
Finding the number of ways of arranging the word ENGINEER such that G and R are always together is done by taking GR as a single alphabet and then finding the permutation. 
Number of ways of arranging the word ENGINEER such that G and R are always together = (7!)/(3!*2!) = 420*2 = 840
∴Number of ways of arranging the word ENGINEER such that G and R are never together = Number of ways of rearranging the word ENGINEER - Number of ways of arranging the word ENGINEER such that G and R are always together 
⇒Number of ways of arranging the word ENGINEER such that G and R are never together 
= 3360 – 840 
= 2520

Q9. Read the following information to answer the following questions : 
Two unbiased dice are thrown simultaneously

What is the probability of getting a doublet?
Answer : Option B
Explaination / Solution:

Total possible outcomes = 6 × 6 = 36 
E = Events of getting a doublet 
= (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) = 6 
PE = 6/36 = 1/6

Q10.
Direction: Study the following information carefully and answer the questions that follow:

A committee of five members is to be formed out of 3 trainees, 4 professors and 6 engineers.
In how many ways this can be done if at least 3 trainees to have to be included in a committee
Answer : Option D
Explaination / Solution:

Number of Ways when if at least 3 trainees include in committee = 
3C3*10C2 = 1*45 --> 45