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 hydraulic structure has four gates which operate independently. The probability of failure of each gate is 0.2. Given that gate I has failed, the probability that both gates 2 and 3 will fail is

  1. 0.240

  2. 0.200

  3. 0.040

  4. 0.008
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The gates operate independently. The probability of failure of each gate is 0.2. Given gate 1 has failed, the probability of gates 2 and 3 failing is P(2 fails) * P(3 fails) = 0.2 * 0.2 = 0.04.

Multiple choice

A box contains 10 screws, 3 of which are defective. Two screws are drawn at random with replacement. The probability that none of the two screws is defective will be

  1. 100%

  2. 50%

  3. 49%

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

Since drawing is done with replacement, the probability of selecting a non-defective screw in a single draw is 7/10. For two independent draws, the probability that neither is defective is (7/10) * (7/10) = 49/100, which is 49%.

Multiple choice

An automobile plant contracted to buy shock absorbers from two suppliers X and Y. X supplies 60% and Y supplies 40% of the shock absorbers. All shock absorbers are subjected to a quality test. The ones that pass the quality test are considered reliable. Of X's shock absorbers, 96% are reliable. Of Y's shock absorbers, 72% are reliable. The probability that a randomly chosen shock absorber, which is found to be reliable, is made by Y is

  1. 0.288

  2. 0.334

  3. 0.667

  4. 0.720
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Multiple choice

In an M/M1 queuing system, the number of arrivals in an interval of length T is a Poisson random variable (i.e the probability of there being n arrivals in an interval of length T is $\frac{e^{-\lambda T(\lambda T)^n}}{n!}$) . The probability density function f(t) of the inter-arrival time is given by
  1. $\lambda^2\bigg(e^{-\lambda^2t}\bigg)$
  2. $\frac{e^{-\lambda^2t}}{\lambda^2}$
  3. $\lambda e^{-\lambda t}$
  4. $\frac{e^{-\lambda^2t}}{\lambda}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Multiple choice statistics normal distribution distribution of measurement probability distributions introduction to normal distribution

The scores on standardized admissions test are normally distributed with a mean of $500$ and a standard deviation of $100$. What is the probability that a randomly selected student will score between $400$ and $600$ on the test?

  1. About $63\%$
  2. About $65\%$
  3. About $68\%$
  4. About $70\%$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
For normal distribution $ P(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)}$
The scores on standardised admissions test are normally distributed with a mean of 500

Mean ($ \mu $)=500

standard deviation($\sigma $)=100
Probability that score lies between 400 and 600  i.e, P(400<x<600)
For standard normal distribution curve
Z=(x-$ \mu $)/$\sigma $
(400-500)/100<(x-$ \mu $)/$\sigma $<(600-500)/100
-1<Z<1
For standard normal distribution mean shifted to zero, $ P(x)dx=\frac{1}{\sqrt{2\pi}}e^{-Z^2/2}dz$ 
Also,P(-1<Z<1)=area of the region between -1 to 1 that is approximately equal to=68%
Multiple choice statistics normal distribution distribution of measurement probability distributions introduction to normal distribution

The length of similar components produced by a company is approximated by a normal distribution model with a mean of $5$ cm and a standard deviation of $0.02$ cm. If a component is chosen at random, what is the probability that the length of this component is between $4.96$ and $5.04$ cm?

  1. $0.9544$
  2. $0.1236$
  3. $0.7265$
  4. $0.9546$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let $x$ be the length of the component.

$x$ has $ \mu=5, \sigma=0.02$

We need to find the probability of the length of the component between $4.96$ and $5.04$. That is to find $P(4.96<x<5.04)$.

Given $x,z=\dfrac{x-\mu}{\sigma}$

Thus for $x=4.96,z=\dfrac{4.96-5}{0.02}=-2$

and for $x=5.04,z=\dfrac{5.04-5}{0.02}=2$

Therefore $P(4.96<x<5.04)=P(-2<z<2)$

                                                       $=P(z<2)-P(z<-2)$

                                                       $=0.9772-0.0228$ (from normal distribution table)

                                                       $=0.9544$

$ \therefore P(4.96<x<5.04)=0.9544$

Hence the probability of the length of the component between $4.96$ and $5.04$ is $0.9544$

Multiple choice maths probability - iii theorem of total probability bayes theorem probability and probability distribution

Which of the following is true regarding law of total probability?

  1. It is a fundamental rule relating marginal probabilities to conditional probabilities.

  2. It expresses the total probability of an outcome which can be realized via several distinct events

  3. Both are correct

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Law of total probability states that,
        If $B _1,B _2,B _3,...$ is a partition of the sample space $S$, then for any event $A$ we have
$P(A)=\sum _iP(A\cap B _i)=\sum _i P(A|B _i)P(B _i)$

That is, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct events and hence the name.

Thus option $(C)$ is correct.