Mathematics

Probability Distributions

457 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

A program consists of two modules executed sequentially. Let $f_1(t)$ and $f_2(t)$ respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by

  1. $f_1(t)+f_2(t)$
  2. $\int_0^t f_1(x)f_2(x)dx$
  3. $\int_0^t f_1(x)f_2(t-x)dx$
  4. $\max\{f_1(t),f_2(t)\}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Probability density function of f1 (t) and f2 (t) is = $\int_0^t f_1(x)f_2(t-x)$

Multiple choice

For each element in a set of size 2n, an unbiased coin is tossed. The 2n coin tosses are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is:
  1. $\frac{^{2n}\mathrm{C}_n}{4^n}$
  2. $\frac{^{2n}\mathrm{C}_n}{2^n}$
  3. $\frac{1}{^{2n}\mathrm{C}_n}$
  4. $\frac{1}{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Required favourable cases = Number of ways of selecting n element out of the set of size 2n = $^{2n}\mathrm{C}_n$ As each toss may result in either head or tail, so for 2n tosses, total no. of combinations = 22n = 4n $\therefore$Required probability = $\dfrac{\text{Favourable cases}}{\text{Total cases}}$ $ = \frac{^{2n}\mathrm{C}_n}{4^n}$

Multiple choice

Aishwarya studies either computer science or mathematics everyday. If she studies computer science on a day, then the probability that the studies mathematics the next day is 0.6. If she studies mathematics on a day, then the probability that the studies computer science the next day is 0.4. Given that Aishwarya studies computer science on Monday, what is the probability that she studies computer science on Wednesday?

  1. 0.24

  2. 0.36

  3. 0.4

  4. 0.6
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The required probability = 0.4 x 0.4 + 0.6 x 0.4 = 0.16 + 0.24 = 0.40

Multiple choice

Consider a company that assembles computers. The probability of a faulty assembly of any computer is p. The company therefore subjects each computer to a testing process. This testing process gives the correct result for any computer with a probability of q. What is the probability of a computer being declared faulty?

  1. pq + (1 − p) (1 − q)

  2. (1 − q)p

  3. (1 − p) q

  4. pq
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A computer is declared faulty in two cases: (1) It is actually faulty AND the test correctly identifies it: probability = p × q. (2) It is NOT faulty BUT the test wrongly declares it faulty: probability = (1-p) × (1-q). These are mutually exclusive events, so we add them: P(declared faulty) = pq + (1-p)(1-q). This is Option A. Option D (pq) only counts true positives, missing false positives. Options B and C only capture one scenario each.