Probability

The post is a quick overview of probability. Probability theory is a branch of mathematics that deals with the occurrence of random events. It provides a framework for quantifying uncertainty and making predictions based on available information.

Overview of Probability

The probability can be classified into two points of view:

• Objective/ Numerical Probability
• Subjective Probability

Objective probability requires a computational formula, while subject probability can be derived from experience, judgment, or previous knowledge about the event. In this post, I will focus on an overview of probability and the Laws of Probability.

Objective Probability

The objective probability has the following definitions

Classical and a Priori Definition

$$P(A) = \frac{\text{Number of sample points in the event based on the favorable events}}{\text{Number of sample points in the sample space}} = \frac{m}{n} = \frac{n(A)}{n(S)}$$

The Relative Frequency or a Posteriori Definition

$$P(A) = \lim\limits_{n\rightarrow \infty} \frac{m}{n}$$

This definition assumes that as $n$ increases to infinity, $\frac{m}{n}$ becomes stable.

When we experiment with the same condition many times, the probability of favourable event becomes stable. For example, if we toss a coin 10 times, then 100 times, then 1000 times, then 10,000 times, then 100,000 times, then 1000,000, and so on. We are also interested in the various numbers of heads that occur. Let $H$ (occurrence of head) be our favorable event, and the probability of a favorable event is called the probability of success. Then the definition said that there are approximately 50% heads in one million tosses. This definition is also called the empirical or statistical definition of probability. that is more useful in practical problems. In practical problems, we find the winning percentage of a team.

The axiomatic Definition of Probability

An axiom is a statement, about any phenomenon, which is used to find real-world problems.

The axiomatic definition of probability states that if a sample space $S$ with sample points $E_1, E_2, \cdots, E_n$, then a real number is assigned to each sample point denoted by $P(E_i)$, should satisfy the following conditions:

• for any event ($E_i$), $0< P(E_i) <1$
• $P(S) = 1$, sure event
• If $A$ and $B$ are two mutually exclusive events, then $P(A\cup B) = P(A) + P(B)$

Laws of Probability

For computing the probability of two or more events, the following laws of probability may be used.

Law of Addition

• For mutually exclusive events: $P(A\cup B) = P(A) + P(B)$
• For non-mutually exclusive events: $P(A\cup B) = P(A) + P(B) – P(A\cap B)$

If $A$, $B$, and $C$ are three events in a sample space $S$, then

$P(A\cup B \cup C) = P(A) + P(B) + P(C) – P(A \cap B) – P(B\cap C) – P(A \cap C)$

Law of Multiplication

For independent events $A$ and $B$: $P(A \text{ and } B) = P(A) \times P(B)$

For dependent events $A$ and $B$: $P(A \text { and } B) = P(A) \times P(B|A)$ (where $P(B|A)$ is the conditional probability of $B$ given $A$)

Law of Complementation

If $A$ is an event and $A’$ is the complement of that event, then

$P(A’) = 1-P(A)$, Note that $P(A) + P(A’) = 1$

Probability of sub-event

If $A$ and $B$ are two events in such a way that $A \subset B$, then $P(A) \le P(B)$

If $A$ and $B$ are any two events defined in a sample space $S$, then

$P(A\cap B’) = P(A) – P(A\cap B)$

Conditional Probability

$P(A|B) = \frac{P(A\cap B}{P(B)}$ or $P(B|A) = \frac{P(A\cap B}{P(A)}$.

Example of Conditional Probability

If we throw a die, what is the probability of 6? That is, $\frac{1}{6}$. What is the probability of 6 given that all are even numbers?

When a die is rolled, the sample space is $S=\{1, 2, 3, 4, 5, 6\}$. Let denote the even numbers by $B$, that is, $B=\{2, 4, 6\}$

$P(A|B) = \frac{1}{3}$

Law of Total Probability

If events $B_1, B_2, \cdots, B_n$ are mutually exclusive and exhaustive events, then for any event $A$: $P(A) = P(A|B_1) \times P(B_1) + P(A|B_2) * P(B_2) + \cdots + P(A|B_n) \times P(B_n)$

Bays’s theorem

Bays’ there is used to update probabilities based on new information.

If $A_1, A_2, \cdots, A_k$ are many events in a sample space.

$P(A_i|B) = \frac{P(A_i) P(B|A_i)}{\Sigma P(A)_i P(B|A_i)}, \text{ for } i, 1, 2, 3, \cdots, k$

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The post is about the Online MCQs Probability Quiz. There are 20 multiple-choice questions covering topics related to random experiments, random variables, expectations, rules of probability, events and types of events, and sample space. Let us start with the Probability Quiz.

Online MCQs Probability Quiz with Answers

• Consider a dice with the property that the probability of a face with $n$ dots showing up is proportional to $n$. What is the probability of the face showing 4 dots?
• Let $X$ be a random variable with a probability distribution function $$f (x) = \begin{cases} 0.2 & \text{for  } |x|<1 \ 0.1 & \text{for } 1 < |x| < 4\ 0 & \text{otherwise} \end{cases}$$ The probability P (0.5 < x < 5) is ————-
• Runs scored by batsmen in 5 one day matches are 50, 70, 82, 93, and 20. The standard deviation is ————-.
• Find the median and mode of the messages received on 9 consecutive days 15, 11, 9, 5, 18, 4, 15, 13, 17.
• $E (XY)=E (X)E (Y)$ if $x$ and $y$ are independent.
• Mode is the value of $x$ where $f(x)$ is a maximum if $X$ is continuous.
• A coin is tossed up 4 times. The probability that tails turn up in 3 cases is ————–.
• If $E$ denotes the expectation the variance of a random variable $X$ is denoted as?
• $X$ is a variate between 0 and 3. The value of $E(X^2)$ is ————-.
• The random variables $X$ and $Y$ have variances of 0.2 and 0.5, respectively. Let $Z= 5X-2Y$. The variance of $Z$ is?
• In a random experiment, observations of a random variable are classified as
• A number of individuals arriving at the boarding counter at an airport is an example of
• If $A$ and $B$ are independent, $P(A) = 0.45$ and $P(B) = 0.20$ then $P(A \cup B)$
• If a fair dice is rolled twice, the probability of getting doublet is
• If a fair coin is tossed 4 times, the probability of getting at least 2 heads is
• If $P(B) \ne 0$ then $P(A|B) = $
• The collection of all possible outcomes of an experiment is called
• An event consisting of one sample point is called
• An event consisting of more than one sample point is called
• When the occurrence of an event does not affect the probability of occurrence of another event, it is called

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

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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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

• 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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