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 business maths probability - iii baye's theorem bayes theorem probability and probability distribution

There are two groups of subjects one of which consists of 5 science subjects and 3 engineering subjects and the other consists of 3 science and 5 engineering subjects. An unbaised die is cast. If number 3 or number 5 turns up, a subject is selected at random from the first group, other wise the subject is selected at random from the second group. Find the probability that an engineering subject is selected ultimately.

  1. $\displaystyle \frac{13}{24}$
  2. $\displaystyle \frac{1}{3}$
  3. $\displaystyle \frac{2}{3}$
  4. $\displaystyle \frac{11}{24}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let  $\displaystyle E _{1}$ be the event that a subject is selected from first group.
$\displaystyle E _{2}$ the event that a subject is selected from the second group.
$E$ be the event that an engineering subject is selected.
Now the probability that die shows $3$ or $5$  is
$\displaystyle P(E _1)=\frac{2}{6}=\frac{1}{3}$

$\displaystyle P\left ( E _{2} \right )=\frac{1}{3}=\frac{2}{3}.$
Now probability of choosing an engineering subject from first group is 
$\displaystyle P\left ( E|E _{1} \right )=$  $\displaystyle \frac{^{3}C _{1}}{^{8}C _{1}}=\frac{3}{8}$
Similarly, $\displaystyle P\left( E|E _{2} \right )=\frac{^{5}C _{1}}{^{8}C _{1}}=\frac{5}{8}$ 
Hence $\displaystyle P\left ( E \right )=P\left ( E _{1} \right )P\left ( E|E _{1} \right )+P\left ( E _{2}\right )P\left ( E|E _{2} \right )$
$\displaystyle =\frac{1}{3}.\frac{3}{8}+\frac{2}{3}.\frac{5}{8}$

$=\dfrac{13}{24}$ 

Multiple choice business maths probability - iii baye's theorem bayes theorem probability and probability distribution

A signal which can be green or red with probability $\displaystyle \frac{4}{5}$ and $\displaystyle \frac{1}{5}$, respectively, is received at station A and then transmitted to station B. The probability of each station receiving the signal correctly is $\displaystyle \frac{3}{4}$. If the signal received at station B is green, then the probability that the original signal was green is

  1. $\displaystyle \frac{3}{5}$
  2. $\displaystyle \frac{6}{7}$
  3. $\displaystyle \frac{20}{23}$
  4. $\displaystyle \frac{9}{20}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
 Event $G$ = original signal is green
$E _1=A$ receives the signal correct
$E _2=B$ receives the signal correct
E = signal received by B is green
$P(\text{signal received by B is green}) = P(GE _1E _2)+ P(G\cap {E _1}\cap {E _2})+ P(\cap GE _1\cap{E _2})+ P(\cap G\cap {E _1}E _2)$
$P(E)=\dfrac {46}{5\times 16}$
$ P(G/E)=\dfrac {\dfrac {40}5\times 16}{\dfrac {46}5\times16}=\dfrac {20}{23}.$
Multiple choice maths frequency distribution tables and graphs introduction to statistics introduction to statistical method and econometrics partition values

In a random number table, each digit appears with___________.

  1. the same frequency

  2. independent of each other

  3. dependent of each other

  4. both (A) and (B)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In a random number table, every digit has got an equal probability of being appearing and are independent of each other as selection of one number is not influenced by any other number.

Multiple choice maths introduction of probability theory types of events some more terms in probability events and its algebra

If the letters of the word  $"ATTEMPT"$  are written down at random. The probability that all the  $T's$  come together is

  1. $1/21$
  2. $6/7$
  3. $1/7$
  4. $1/42$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Number of ways of arranging the words keeping all $T's$ together is  $5!$

Number of ways of arranging the words  is  $\dfrac{7!}{3!}$
Probability that all the $T's$  together is $\dfrac{5! }{\dfrac{7!}{3!}}=\dfrac{1}{7}$

Multiple choice maths introduction of probability theory types of events some more terms in probability events and its algebra

The probability of a certain event is 

  1. $0$
  2. $1$
  3. greater than $1$
  4. less than $0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
An event which always happens is called a sure event or a certain event. So the probability of a certain event is $1$. 
For example, when we throw a die, then the event "getting a number less than $7$" is a certain event.
Multiple choice maths introduction of probability theory types of events some more terms in probability events and its algebra

If P(E) = 0 then E is a/an

  1. sure event

  2. impossible event

  3. equally likely event

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

$P(E)=\frac{number  of   outcomes  favorable}{Total   numbers  of  possible  outcomes}$

If P(E)=0 then the event is called impossible event.
For example -
When a dice is thrown the possible outcomes are 1,2,3,4,5 and 6.
then  the probability is to getting the number 7  in a single throw of a dice is 0 then this is called impossible event.
$P(E)=\frac{0}{6}=0$    

Multiple choice maths introduction of probability theory types of events some more terms in probability events and its algebra

The probability of an event which is sure to occur at every performance of an experiment is called a ___________.

  1. simple event

  2. compound event

  3. complementary event

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

The probability of an event which is sure to occur at every performance of an experiment is called a certain event.
Example: Head or Tail is a certain event connected with tossing a coin.

Multiple choice maths introduction of probability theory types of events some more terms in probability events and its algebra

The probability of a sure event (or certain event) is ____

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

Probability of event in $E$, $ P\left( E \right) =\dfrac { \text {number of events in E} }{ \text {Tota number of possible events} } $

If the number of desirable events is equal to the total number of possible events, then probability of an event which is certain or sure to happen $ = 1$
Such an event is called certaind event or sure event.