Mathematics

Probability Distributions

488 Questions

Probability distributions describe the likelihood of different outcomes in a statistical experiment. Key topics include calculating confidence intervals, understanding Type II errors, and working with the standard normal distribution. These concepts are frequently tested in mathematics and statistics sections of various competitive exams.

Standard normal distributionType II error probabilityConfidence interval interpretationPoisson distributionProbability density function

Probability Distributions Questions

Multiple choice

Students in a class of strength 'p' have roll numbers 1, 2, 3, 4… p - 1, p. Take a student with roll number 'n'. What is the probability that either he or someone with a roll number higher than him is selected at random?

  1. 1/p

  2. (p - n)/p

  3. (p - n + 1)/p

  4. 1/n

  5. n/p
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

There are (p-n) students in the class with roll number greater than 'n'. Add one to this to include the student with roll number 'n'. Hence this gives the probability that the selected student has his roll number 'n' or greater than 'n'. This is the required probability.

Multiple choice

In a family of 5 persons (father, mother, a son and two daughters), 2 are to be selected randomly to go out for dinner. What is the probability that at least one of the daughters is selected for this purpose?

  1. 0.7

  2. 0.4

  3. 0.5

  4. 2/3

  5. 0.2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

2 persons can be selected out of 5 in 5C2 = 10 ways.

Number of selections in which none of the daughters is selected = 3C2 = 3.

In remaining selections (10 – 3 = 7) at least one of the daughters will be present.

So the required probability is, 7/10 = 0.7.

Multiple choice

Four students reached late in a class and made an excuse to the teacher that they had a flat tyre while coming to the college by car. The teacher asked them to sit at four distant seats and made them to answer the same question “which tyre of the car was flat”? What is the probability that all the four give the same answer and their lie is not caught? Assume that students had told lie to the teacher and they will now select their answers randomly.

  1. 1/4

  2. 1/64

  3. 1/256

  4. 1/16

  5. 1/32
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

There are 4 tyres in a car. A person can answer any of the four tyres as flat. Thus, all the four students have four options for giving their answers. There can be 4*4*4*4 = 256 different answers. Out of these, only 4 possibilities are there in which all the four students choose the same tyre as flat. That happens when the first person answers any of the four tyres and remaining three choose the same one. So the required probability is 4/256 = 1/64.

Multiple choice

In a class of 70 students, there are 5 students who are better than others and they have double the chance of being placed at the first rank in a test in comparison to other students. Vijay is not among these five. What is the probability that Vijay gets the first rank in the test?

  1. 0

  2. 1/70

  3. 1/140

  4. 1/75

  5. 1/65
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Since, 5 students have double the chance of being first than others; you may consider each of them as 2 and compute the probability. Thus, these 5 are equivalent to 10 and remaining 65 added to this make it 75. These 5 students will have probability of being first as 2/75 and remaining 65 students will have the same probability as 1/75. Thus 65 times 1/75 and 5 times 2/75 make it 1. Vijay is one among the 65 students. This is correct.

Multiple choice

Two distinct numbers a and b are chosen randomly from the set {2, 22, 23, 24, ------ 225} (25 numbers are there). Find the probability that logab (log b to the base a) is an integer.

  1. 2/25

  2. 31/150

  3. 31/300

  4. 37/300

  5. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For Logab to be integer, b>a. So, obviously b cannot be 2 because a and b are distinct. So, if b=2, a should be greater than that, and then it will not be an integer. So if a=2 ,then possible values of b are {22,23,24,------224,225} There are 24 possible values. Similarly, if a=22, then possible values of b are {24,26,28,------222,224}, i.e. there are 11 possible values. Similarly, if a=23, then possible values of b are {26,29,212,------221,224}, i.e. there are 7 possible values. Similarly, if a=24, then possible values of b are {28,212,216,------220,224}, i.e. there are 5 possible values. Similarly, if a=25, then possible values of b are {210,215,220,------225}, i.e. there are 4 possible values. Similarly, if a=26,then possible values of b are {212,218,224}, i.e. there are 3 possible values. Similalry, if a=27, then possible values of b are {214,221}, i.e. there are 2 possible values. Similalry, if a=28, then possible values of b are {216,224}, i.e. there are 2 possible values. Similarly, if a=29, then possible values of b are {218), i.e. only 1 possible value. For a=210,211,212, the possible values are 220,222,224 respectively. After that, there is no case possible. So, total possibilities are (24+11+7+5+4+3+2+2+1+1+1+1)= 62 cases Total number of possibilities = a can be 25 numbers and b can be 24 numbers (As a and b should be distinct) Total possibilities are 25*24. Probability = 62/(25*24) = 31/300 (Correct Answer)

Multiple choice
If one number is chosen at random from the first 900 positive integers, what is the probability that the number chosen is a multiple of both 3 and 5?

  1. 3/125

  2. 1/9

  3. 3/45

  4. 9/75
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Any number that is a multiple of both 3 and 5 must be a multiple of 15. The number of integers from 1 and 900 which are multiples of 15 are 15 × 1, 15 × 2, 15 × 3, ..., 15 × 60. These are 60 in number. So, the probability that one among the first 900 numbers chosen at random being a multiple of 15 is 60/900 = 6/90 = 1/15 = 3/45

Multiple choice

A box contains 20 electric bulbs, out of which 5 are defective. Two bulbs are chosen at random from this box. The probability that at least one of these is defective is

  1. 1/4

  2. 17/38

  3. 21/38

  4. 3/38

  5. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

P (none of the balls is defective) = 15C2/20C2

                                                         = 21/38

P(at least one ball is defective) = 1 - 21/38

                                                    = 17/38

 

Multiple choice

A candidate is to be selected for an interview for 2 posts. The number of candidates for the first post is 5 and for the second post is 6. Find the probability of getting at least one job.

  1. 5/11

  2. 6/11

  3. 2/11

  4. 1/3

  5. 1/11
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The calculation of probability is as follows:
 
Probability that the candidate does not get an offer from the first interview is 4/5.
Probability that the candidate does not get an offer from the second interview is 5/6.
Total probability of not getting an offer is 4/5 * 5/6 = 2/3
Hence, the probability of getting at least one job is 1 - 2/3 = 1/3  
 
 

Multiple choice

A hash table has space for 100 records. What is the probability of collision before the table is 10% full?

  1. 0.45

  2. 0.5

  3. 0.3

  4. 0.34
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

This is a variation of the birthday paradox. The probability of no collision for n items in m slots is (m/m) * ((m-1)/m) * ... * ((m-n+1)/m). For 10 records in 100 slots, the probability of no collision is approx 0.63, meaning the probability of at least one collision is 1 - 0.63 = 0.37, which is closest to 0.34.