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 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 bio-chemistry inborn errors of metabolism haemophilia and colour blindness non infectious diseases keeping diseases away

If both parents are bald and their first female child is normal, then the chances of baldness in their second male child is?

  1. $50$%
  2. $100$%
  3. $25$%
  4. $0$%
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Male pattern baldness is a sex-influenced trait. In males, it acts as a dominant trait, while in females, it acts as a recessive trait. If both parents are bald, the male child has a 100% chance of inheriting the allele and expressing the phenotype.

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

The term law of total probability is sometimes taken to mean the ____

  1. Law of total expectation

  2. Law of alternatives

  3. Law of variance

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

The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables.

Hence, option B is correct.

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

The events $E _1, E _2, ........$ represents the partition of the sample space $S$, if they are:

  1. pairwise disjoint

  2. exhaustive

  3. have non-zero probabilities

  4. All are correct
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A set of events $E _1 , E _2 ,...$  is said to represent a partition of a sample space $S$, if 


$(a)$  $E _i \cap E _j = \phi, i\neq j; i, j = 1, 2, 3,..., n$  (pairwise disjoint)

$(b)$ $E _i \cup E _2 \cup ... \cup E _n = S$ (exhaustive)

$(c)$ Each $E _i \neq \phi, i.e, P(E _i) > 0$ for all $i = 1, 2, ..., n$ (have non-zero probabilities)

That is the events should be pairwise disjoint, exhaustive and should have non zero probabilities.

Hence, option D is correct.

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

In a construction job, following are some probabilities given:
Probability that there will be strike is $0.65$, probability that the job will be completed on time if there is no strike is $0.80$, probability that the job will be completed on time if there is strike is $0.32$. Determine probability that the construction job will get complete on time.

  1. $0.438$
  2. $0.538$
  3. $0.488$

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

Let $A$ be the event that the construction job is completed on time and $B$ be the event that there is strike.


$\Rightarrow P(B)=0.65$

Hence probability that there will be no strike $=P(B')$
                                                                            $=1-P(B)$
                                                                            $=1-0.65$
                                                                            $=0.35$

$\therefore P(B')=0.35$

By the Law of Total Probability we have $P(A)=P(B) \times P(A|B)+P(B') \times P(A|B')$

Given, $P(construction : job: is: completed: with: no: strike)=P(A|B)=0.80$ 
and $P(construction : job: is: completed: with:  strike)=P(A|B')=0.32$

$\therefore P(A)=0.65 \times 0.80+0.35 \times 0.32=0.488$

Hence the probability that the construction job will get complete on time is $0.488$

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

There are three boxes, each containing a different number of light bulbs. The first box has 10 bulbs, of which four are dead, the second has six bulbs, of which one is dead, and the third box has eight bulbs of which three are dead. What is the probability of a dead bulb being selected when a bulb is chosen at random from one of the three boxes?

  1. $\dfrac{115}{330}$
  2. $\dfrac{113}{360}$
  3. $\dfrac{113}{330}$

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

Let $𝐴 _1, 𝐴 _2, 𝐴 _3$ denotes the events of selecting bulbs from bags $1,2: and: 3$ respectively. 


Let $𝐵$ denotes the event the bulbs selected are dead.

$ 𝑃(𝐴 _1) = 𝑃(𝐴 _2)  = 𝑃(𝐴 _3)  = \dfrac{1}{3} $

Also $P(B|A _1)=\dfrac{4}{10}, P(B|A _2)=\dfrac{1}{6}, P(B|A _3)=\dfrac{3}{8}$

By law of total probability,

$P(B)=P(A _1)P(B|A _1)+P(A _2)P(B|A _2)+P(A _3)P(B|A _3)$

Substituting the values we get,

$P(B)=\dfrac{1}{3} \times \dfrac{4}{10}+ \dfrac{1}{3} \times \dfrac{1}{6}+ \dfrac{1}{3} \times \dfrac{3}{8}$

$\Rightarrow P(B)=\dfrac{113}{360}$

Thus the probability of a dead bulb being selected when a bulb is chosen at random from one of the three boxes is $\dfrac{113}{360}$.


Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method for calculating mean deviation about mean mean deviation about mean and median range and mean deviation

Probability is expressed as _______.

  1. percentage

  2. ratio

  3. proportion

  4. all (a), (b), (c)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Probability shows the relationship between two variables in the form of ratio, percentage or proportion where there the chances of occurrence of one variable is expressed in terms of other variable. Since the value of one variable belongs to the range of value of another variable, the range o probability varies from 0 to 1.
Multiple choice biology mendel's law of inheritance determination of sex sex determination sex determination in humans

If the first seven children born to a particular pair of parents are all males, what is the probability that the eighth child will also be a male?

  1. $\dfrac{1}{2}$
  2. $\dfrac{1}{4}$
  3. $\dfrac{1}{8}$
  4. $\dfrac{1}{16}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
The sex of an individual depends on the combination of sex chromosomes in its cells, that is, the combination of X and Y chromosomes. An embryo with XX genotype would develop into a female, while an embryo with an XY would develop into a male. The mother has a genotype and would always produce gametes with an X chromosome. The father, however, would produce half the gametes having an X chromosome and half carrying a Y chromosome. The sex of the embryo in humans would hence depend on the chromosome present in the gamete of the father. For every child conceived, there is an equal probability of the child being a male or a female, as the gamete received from the father has a 50% chance of carrying an X chromosome and  50% chance of carrying a Y chromosome.
So, the correct answer is $\dfrac{1}{2}$
Multiple choice botany mendel's law of inheritance genetics terminology similarities and dissimilarities in organisms introduction to genetics features of human body and sign language

A husband and wife have normal vision but fathers of both of them were colour blind. Probability of their first daughter to be colour blind is

  1. 25%

  2. 50%

  3. 75%

  4. 0%
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Probability of their first (or any) daughter being colourblind is nil as the daughter of normal father can never be colourblind.