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

An examination consists of two papers, Paper 1 and Paper 2. The probability of failing in Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed in Paper 2, the probability of failing in Paper 1 is 0.6. The probability of a student failing in both the papers is

  1. 0.5

  2. 0.18

  3. 0.12

  4. 0.06
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\text{Probability of failing in paper 1 is} \hspace{1cm} P(A) = 0.3 \\ \text{Possibility of failing in paper 2 is} \hspace{1cm} P(B) = 0.2 \\ Probability of failing in paper 1, when \\ \text{Student has failed in paper 2 is} \hspace{1cm} P(\frac{A}{B}) = 0.6 \\ \text{We know that} \\ P(\frac{A}{B}) = \frac{(P \cap B)}{P(B)} \\ or \hspace{1cm} P(A \cap B) = P(B) P (\frac{A}{B}) = 0.6 \times 0.2 = 0.12$

Multiple choice

Px (x) = M exp (- 2 |x|) - Nexp (- 3 |x|) is the probability density function for the real random variable X, over the entire x axis, M and N are both positive real numbers. The equation relating M and N is
  1. M - $\frac{2}{3}$N = 1
  2. 2M + $\frac{1}{3}$N = 1

  3. M + N = 1

  4. M + N = 3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a valid PDF over entire x-axis: ∫{-∞}^{∞} p_x(x)dx = 1. With even function: 2∫{0}^{∞}[Me^{-2x} - Ne^{-3x}]dx = 1. Integrating: 2[M/2 - N/3] = 1, so M - N/3 = 1, giving M - (2/3)N = 1. Note: the expression in option A correctly represents this constraint.

Multiple choice

A kid is carrying 15 identical but differently colored crayons in his pocket, out of these 5 are blue, 7 green and 3 yellow. If he takes out two crayons randomly from his pocket, what is the probability that they are green and yellow?

  1. 1/5

  2. 2/5

  3. 3/5

  4. 4/5

  5. 1
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

2 crayons can be drawn out of 15 in 15C2 = 105 ways.  One green crayon and one yellow crayon separately can be drawn in 7C1 = 7 and 3C1 = 3 ways, respectively. Since they are exclusive and independent events, taking one green and one yellow crayon together will make the favourable number of ways equal to 7C1 x 3C1 = 21 ways. Hence, required probability, P = 7C1 x 3C1/(15C2) = 21/105 = 1/5

Multiple choice

A shopkeeper received 3 differently sized vests to be delivered to 3 different customers. His helper delivered them randomly to the customers without worrying about their respective sizes. What is the probability that at least one of them received the right sized vest?

  1. 1/3

  2. 2/3

  3. 1

  4. 3/4

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

Let the order of correct sizes be (1, 2, 3). The vests can be delivered to the customers in the following ways: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2) and (3, 2,1) = 6 ways. The cases (1, 2, 3), (1, 3, 2), (2, 1, 3) and (3, 2, 1) correspond to at least one right sized delivery OR 4 ways. Therefore, the required probability = 4/6 = 2/3

Multiple choice

If a student is picked randomly from a class of 40 students, what is the probability that his name starts with a vowel? Assume that there is no bias towards name starting with any letter.

  1. 5/40

  2. 1/40

  3. 5/26

  4. 5/21

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

Since there is no bias towards any letter, this is the right answer because there are 5 vowels out of 26 letters in the English.

The number of students is not a multiple of 26 but still the exact probability will be close to this.

Multiple choice

A couple has two children. What is the probability that they have one son and one daughter? Assume that there is no bias towards son or daughter.

  1. 1/3

  2. 1/2

  3. 1/4

  4. 1

  5. It depends on other factors
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

First issue can be a son or a daughter with probabilities of 1/2. Similarly the second issue too can be a son or a daughter with equal probabilities of 1/2.

So, there are four equally probable possibilities out of which two result in one son - one daughter combination. Required probability should be 2/4 or 1/2.