Muhammad Imdad Ullah

Currently working as Assistant Professor of Statistics in Ghazi University, Dera Ghazi Khan. Completed my Ph.D. in Statistics from the Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan. l like Applied Statistics, Mathematics, and Statistical Computing. Statistical and Mathematical software used is SAS, STATA, Python, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel. Like to use type-setting LaTeX for composing Articles, thesis, etc.

Counting Techniques in Probability Statistics

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The counting techniques in probability, statistics, mathematics, engineering, and computer science are essential tools. Counting Techniques in probability help in determining the number of ways a particular event can occur.

The following are the most common counting techniques in probability theory:

Factorial

For any integer $n$, $n$ factorial (denoted by $n!$) is the descending product beginning with $n$ and ending with 1. It can be written as

$$n! = n\times (n-1) \times (n-2) \times \cdots \times 2 \times 1$$

The example of factorial counting are:

  • $3! = 3\times 2\times 1 = 6$
  • $5! = 5\times 4\times 3! = 20 \times 6 = 120$
  • $10! = 10\times 9\times 8\times 7\times 6\times 5! = 3628800$

Note that a special definition is made for the case of $0!$, $0!=1$.

Permutations

A permutation of a group of objects is an ordered arrangement of the objects. The number of different permutations of a group of $n$ objects is $n!$. The formula of permutation is

$$P(n, r) = {}^nP_r = \binom{n}{r} = \frac{n!}{(n-r)!}$$

where $n$ is the total number of objects, and $r$ is the number of objects to be arranged.

The example of permutations are:

  • The number of ways of dealing with the cards of a standard deck in some order is $52! = 8.066\times 10^{67}$
  • Suppose, we want to place a set of five names in some order, there are five choices for which name to place first, then 4 choices of which to list second, 3 choices for third, 2 choices for fourth, and only one choice for the last (fifth one). Therefore, one can determine, how many different ways can 5 people be ordered in a row can be counted using the fundamental counting principle, the number of different ways to put 5 names in order is $5! = 5\times 4 \times 3\times 2\times 1 = 120$

Often entire set of objects is not required to be placed in order, usually one wants to compute how many ways a few chosen objects can be ordered. For example,

Example: A horse race has 14 horses, how many different possible ways can the top 3 horses finish?
Solution: There are 14 possibilities for which horse finishes first, 13 for second, and 12 for third. So, by the fundamental counting principle, there are $14\times 13\times 12 = 2184$ different possible ways (finishing orders) for the top three horses. $\binom{14}{3} = \frac{14!}{(14-3)!}=2184$.

In the above examples, permutations are called permutations of $n$ objects taken $r$ at a time.

Combinations

A combination is a selection of objects from a set without regard to order. Combinations are used when calculating the number of outcomes for experiments involving multiple choices, and often the order of the choices is not required. The formula of combination is

$$C(n, r) = {}^nC_r = \frac{n!}{r!(n-r)!}=\frac{{}^nP_r}{r!}$$

where $n$ is the total number of objects, and $r$ is the number of objects to be arranged without any regard to importance or order.

The example of combinations are:

  • Drawing a 5-card poker hand (${}^{52}C_5$)
  • Selecting a three-person committee from a group of 30 (without any priority or importance) (${}^{30}C_3$)

A choice of $r$ objects from a group of $n$ objects without regard to order is called a combination of $n$ objects taken $r$ at a time.

Example: In how many different ways can a committee of 3 people be chosen from a group of 10 people?

Solution: $C(10, 3) = \frac{10!}{3!(10-3)!} = 120 ways$

Counting Techniques in Probability

Multiplication Principle

If one event can occur $m$ times and another event occurs $n$ times, then the occurrence of the two events together can be computed using the multiplication principle, that is, by multiplying $m\times n$. For example, if there are 5 shirts and 3 pants to choose from, one can compute the different ways of outfits by multiplying the number of shirts and number of pants, i.e., $5 \times 3=15$, so there are 15 ways of outfits from 5 shirts and 3 pants.

Addition Principle

If one event can occur in $m$ ways and a second event can occur n $n$ ways, then one or the other event can occur in $m+n$ ways. For example, if there are 3 red balls and 4 blue balls, the number of was a ball can be chosen is: $4+3=7$.

Application of Counting Techniques in Probability

  • Probability: Calculating probabilities of events based on the number of favorable outcomes and the total number of possible outcomes.
  • Combinatorics: Studying the arrangement, combination, or selection of objects.
  • Computer Science: Analyzing algorithms and data structures.
  • Statistics: Sampling and hypothesis testing.
  • Cryptography: Designing secure encryption methods

FAQs about Counting Techniques in Probability

  1. What is meant by counting techniques?
  2. What are the applications of counting techniques in probability?
  3. Define permutations and combinations.
  4. What is the difference between the multiplication and addition principles?
  5. Give real-life examples of permutations and combinations.
  6. Write down the formulas of permutations and combinations.
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Basic Statistics MCQs with Answers 15

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This post is about Basic Statistics MCQs with Answers. There are 20 multiple-choice questions from the construction of frequency distribution, cumulative frequency, class intervals, class boundaries, and class width. Let us start with Basic Statistics MCQs with Answers.

Multiple-Choice Questions about Frequency Distribution Table

1. General tables of data used to show data in an orderly manner are called as

 
 
 
 

2. A complex type of table in which variables to be studied are subdivided with interrelated characteristics is called as

 
 
 
 

3. The exclusive method and inclusive method are ways of classifying data on the basis of

 
 
 
 

4. Table in which data represented is extracted from some other data table is classified as

 
 
 
 

5. Which one of the following is the class frequency?

 
 
 
 

6. The ‘less than type distribution’ and ‘more than type distribution’ are types of

 
 
 
 

7. A term used to describe frequency curve is

 
 
 
 

8. The class interval classification method which ensures data continuity is classified as

 
 
 
 

9. Frequencies of all specific values of x and y variables with total calculated frequencies are classified as

 
 
 
 

10. Distribution which shows a cumulative figure of all observations placed below the upper limit of classes in distribution is considered as

 
 
 
 

11. The type of table in which study variables provide a large number of information with interrelated characteristics is classified as

 
 
 
 

12. Frequency distribution which is the result of cross-classification is called

 
 
 
 

13. ‘less than type’ cumulative frequency distribution is considered as correspondence to

 
 
 
 

14. Simple classification and manifold classification are types of

 
 
 
 

15. A distribution which requires the inclusion of open-ended classes is considered as

 
 
 
 

16. The classification method in which the upper and lower limits of the interval are also in the class interval itself is called

 
 
 
 

17. The type of classification in which a class is subdivided into subclasses and subclasses are divided into more classes is considered as

 
 
 
 

18. The type of cumulative frequency distribution in which class intervals are added in bottom-to-top order is classified as

 
 
 
 

19. Cumulative frequency distribution which is the ‘greater than’ type is correspondent to

 
 
 
 

20. The type of classification in which a class is subdivided into subclasses and one attribute is assigned for statistical study is considered as

 
 
 
 

Basic Statistics MCQs with Answers

Online Basic Statistics MCQs with Answers
  • The classification method in which the upper and lower limits of the interval are also in the class interval itself is called
  • General tables of data used to show data in an orderly manner are called as
  • Frequencies of all specific values of x and y variables with total calculated frequencies are classified as
  • A term used to describe frequency curve is
  • Distribution which shows a cumulative figure of all observations placed below the upper limit of classes in distribution is considered as
  • A distribution which requires the inclusion of open-ended classes is considered as
  • The type of cumulative frequency distribution in which class intervals are added in bottom-to-top order is classified as
  • The ‘less than type distribution’ and ‘more than type distribution’ are types of
  • The exclusive method and inclusive method are ways of classifying data on the basis of
  • The type of classification in which a class is subdivided into subclasses and subclasses are divided into more classes is considered as
  • Frequency distribution which is the result of cross-classification is called
  • The type of table in which study variables provide a large number of information with interrelated characteristics is classified as
  • Table in which data represented is extracted from some other data table is classified as
  • The class interval classification method which ensures data continuity is classified as
  • Which one of the following is the class frequency?
  • A complex type of table in which variables to be studied are subdivided with interrelated characteristics is called as
  • ‘less than type’ cumulative frequency distribution is considered as correspondence to
  • The type of classification in which a class is subdivided into subclasses and one attribute is assigned for statistical study is considered as
  • Cumulative frequency distribution which is the ‘greater than’ type is correspondent to
  • Simple classification and manifold classification are types of
Basic Statistics MCQs with Answers 15

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Binomial Random Variables

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Introduction

A discrete random variable $x$ is said to have a binomial distribution if $x$ (binomial random variable) satisfies the following conditions:

  • An experiment is repeated for a fixed number of trials $n$.
  • All the trials of the experiments are independent of each other.
  • All possible outcomes for each trial of the experiment can be classified into two mutually (complementary) events: one is $S$ (called success) and the other is $F$ (called failure).
  • The probability of success $P(S)$ has a constant value of $p$ for every trial (that is, the probability of success is fixed for each trial) and hence the probability of failure $P(F)$ has a constant/fixed value of $q$ for every trial, where $q=1-p$.
  • The random variable $x$ counts the number of trials on which $S$ (success) occurred.

Calculating Probabilities for a Binomial Random Variable

If $X$ is a binomial random variable with $n$ trials, probability of success $p$ (and probability of failure $q$), then by the fundamental counting principle, the probability of any outcome in which there are $x$ successes (and therefore $n-x$ failures) is

Binomial random variables

To count the number of outcomes with $x$ successes and $n-x$ failures, one can observe that the $x$ successes could occur on any $x$ of the $n$ trials. The number of ways of choosing/selecting $x$ trials out of $n$ is $\binom{n}{x}$, so the probability of $x$ successes becomes:

$$P(X=x)=\binom{n}{x} p^x q^{n-x}$$

Example of Binomial Random Experiments

Example: Consider the experiment of flipping a coin 5 times. Let the event of getting Tails on a flip is considered a “success”. Also, suppose that the random variable $T$ is the number of tails obtained, the $T$ will be binomially distribution with $n=5, p=\frac{1}{2}$, and $q=\frac{1}{2}$.

Solution:
Suppose the random variable $T$ represents the number of trials when a coin is flipped three times.
$$P(X=2) = \binom{3}{2}\left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^1 = 0.375$$

Properties of Binomial Distributions

In many cases, one may be interested in the mean and standard deviation of the binomial random variable. If $x$ is a binomial random variable with $n$ trials with probability of success $p$ and probability of failure $q$, then the mean and standard deviation of $x$ can be computed as

  • Mean: $E(X) = \mu(x) = np$
  • Standard Deviation: $\sigma(x) = \sqrt{npq}$
  • Variance: $npq$

Note that

  • A binomial distribution is symmetric if $p=q$,
  • left skewed if $p>q$ and
  • right-skewed if $p<q$

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Important Probability MCQs Online Test 9

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The post is about the Probability MCQS Online Test. There are 20 multiple-choice questions covering the basics of probability, events and types of events, rules of probability, subjective and classical probability, etc. Let us start with the Probability MCQs Online Test.

Please go to Important Probability MCQs Online Test 9 to view the test

Probability MCQs Online Test

Probability MCQs Online Test
  • A die is rolled. What is the probability that the number rolled is greater than 2 and even
  • If you have two standard decks of cards (one with a blue back and the other with a red back), you draw one card from each. What is the probability that the blue-backed card is a heart or the red-backed card is a black card?
  • If you roll a pair of dice, what is the probability that (at least) one dice is a 1 or the sum of the dice is 9?
  • If a card is chosen from a standard deck of cards, what is the probability of getting a diamond or a club?
  • If a card is chosen from a standard deck of cards, what is the probability of getting a two or a five?
  • If a card is chosen from a standard deck of cards, what is the probability of getting a five or a face card?
  • If a letter is chosen at random from the 10 letters of the word STATISTICS, what is the probability that it is a vowel?
  • From the following table, what is the probability of selecting a female university graduate student from this group?
  • Subjective probabilities are assigned to the events $A$ and $B$, which comprise a sample space. Which of the following probability statements is not valid?
  • A personal manager selects an applicant at random from a large group for an interview. The probability of the applicant being male is 0.60. The probability of selecting an adult is 0.70. The probability of selecting an adult male is 0.45. Given that a male is selected, the probability that he is an adult is:
  • Which of the following is a collection of all mutually exclusive events representing a card randomly selected from a deck of ordinary playing cards?
  • Which of the following are collectively exhaustive events representing a card randomly selected from a deck of ordinary playing cards?
  • Indicate in which one of the following situations the events $A$ and $B$ are independent:
  • Which one of the following statements is false?
  • A marginal probability might be found by any but which one of the following?
  • Which one of the following statements is not true?
  • If $A$ and $B$ are independent events, $P(A) = 0.45, P(B) = 0.60$ then $P(A \cap B)$ is
  • The probability of drawing one white ball from a bag containing 2 white, 3 blue, and 3 black balls is
  • A letter is chosen at random from the word MATHEMATICS, the probability of getting $M$ is
  • If $P(B|A) = P(B)$ then $A$ and $B$ are
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