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An Introduction to Probability and Statistics

An Introduction to Probability and Statistics

Probability and statistics are two key concepts in mathematics. Probability is based entirely on chance. Statistics, on the other hand, is involved with how we handle various data sets using various techniques. It aids in the representation of complex data in a simple and understandable manner. Nowadays, statistics are widely used in data science professions. Professionals use statistics to make business predictions. It assists them in forecasting the company's future profit or loss.


Important Topics of Statistics and Probability

  • Probability and Statistics

  • Probability

  • Statistics

  • Probability Distribution

  • Types of Events in Probability

  • Types of Data in Statistics

  • Mean, Median and Mode

  • Mean Median Mode Formula

  • Types of statistics

 

Important Concepts of Statistics and Probability

Definition of Probability

Probability refers to the possibility of any random event's outcome. For instance, when an event occurs, such as throwing a ball or selecting a card from a deck, there must be some probability associated with that event. This term refers to determining the probability of a given event occurring. 

For example, when we flip a coin in the air what are the chances of getting a head?

The number of possible outcomes determines the answer to this question. In this case, the outcome could be either head or tail. As a result, the probability of a head appearing as a result is $\dfrac{1}{2}$.


Probability Formulas

Any event that occurs has an outcome. An event can have a variety of outcomes.

The formula for calculating probability is given by,

${P(\text{Event})} = \dfrac{\text{Total Number of Favorable Outcomes}}{\text{Total number of outcomes}}$

$P(E) = \dfrac{n(E)}{n(S)}$

Where, 

n(E) - the outcome of an event

n(S) - the total number of events in the sample space


Probability Formula for Two Events

Let A and B be two events

Probabilities

Definition

Formula

Probability Range

An event's probability ranges from 0 to 1.

$0 \leq P(A) \leq 1$

Complementary Events Rule

One event occurs if and only if the other does not, the two events are said to be complementary.

$P(A^\prime) + P(A) = 1$

Addition Rule

The probability of two mutually exclusive events or two non-mutually exclusive events occurring.

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

Multiplication Rule

The product of the probability of B occurring and the conditional probability of event A occurring given that event B occurs is the probability of both events A and B occurring.

$P(A\cap B) = P(B) \cdot P(A|B)$

Mutually Exclusive Events

No two or more events can occur simultaneously at the same time.

$P(A\cup B) = P(A) + P(B)$

Independent Events

Those events whose occurrence is unaffected by any other events.

$P(A\cap B) = P(A)P(B)$

Disjoint Events

They have no common outcome.

$P(A\cap B) = 0$

Conditional Probability

The probability of an event or outcome occurring based on the occurrence of a previous event or outcome.

$P(A|B) = \dfrac{P(A\cap B)}{P(B)}$

Bayes Formula

The probability of an event occurring in relation to any condition is described by Bayes' theorem.

$P(A|B) = \dfrac{P(B|A) \cdot P(A)}{P(B)}$


What Are the Five Probability Rules?

  • The probability of an impossible event is phi or a null set.

  • The sample space of an event contains the maximum probability of that event (sample space is the total number of possible outcomes).

  • Any event has a probability between 0 and 1 but an event cannot have a negative probability.

  • If A and B are two events that cannot occur simultaneously, then the probability of A or B occurring is the probability of A + the probability of B.


What is Statistics?

The study of data collection, analysis, interpretation, presentation, and organization is known as statistics. It is a method of gathering and summarizing data. Stats are used for all data analysis, whether it is the study of the country's population or its economy.


Statistics has a wide range of applications in many fields, including sociology, psychology, geology, weather forecasting, and so on. The information gathered for analysis here could be quantitative or qualitative. Quantitative data can also be divided into two types: discrete and continuous. Continuous data has a range rather than a fixed value, whereas discrete data has a fixed value. There are many topics in statistics, and terms in statistics and formulas used in this concept.


Types of Statistics

There are two types of statistics in general.

Descriptive Statistics: 

The data is summarized and described in descriptive statistics. The summarization is done from a population sample using various parameters such as mean and standard deviation. Descriptive statistics are a way of organizing, representing, and explaining a set of data using charts, graphs, and summary measures.

Histograms, pie charts, bars, and scatter plots are common ways to summarize data and display it in tables or graphs.


Inferential Statistics: 

We use inferential statistics to describe the meaning of the collected data after it has been collected, analyzed, and summarized.

The probability principle is used in inferential statistics to determine whether trends found in a research sample can be generalized to the larger population from which the sample was drawn.

Inferential statistics are used to test hypotheses and investigate relationships between variables, and they can also be used to predict population size.


Formula for Statistics

In statistics, there are five key formulas. They are:

1. $\text{Mean}=\bar{x}=\dfrac{\sum x_{i}}{N}$

Where, $x_{i}$ - terms in the data set

$N$ - the total number of terms

2. If ‘n’ is odd number, $\text{Median}=\left(\dfrac{n+1}{2}\right)^{th}$

If ‘n’ is an even number, $\text{Median} = \dfrac{1}{2}\left\{\left(\dfrac{n}{2}\right)^{th}\text{ variable } + \left(\dfrac{n}{2} + 1\right)^{th}\text{ variable}\right\}$

3. The most frequently occurring value in a data set is referred to as the mode.

4. Variance $=\sigma^{2}=\dfrac{\sum\left(x_{i}-x\right)^{2}}{N}$

5. Standard Deviation $=\sqrt{\text { Variance }}$

Or, $\sqrt{\dfrac{\sum\left(x_{i}-x\right)^{2}}{N}}$


Measures of Central Value

Sometimes when we are working with large amounts of data, we need one single value to represent the whole set. In math, there are three measures to find the central value of a given set of data. They are


Mean

The term mean refers to the average of a set of numbers.

$\text{Mean}=\dfrac{\text{Sum of all the numbers}}{\text{Total number of items}}$

Mean Formula is given by 

$\text{Mean} = \dfrac{\sum(f_i \cdot x_i)}{\sum f_i}$


Steps to Calculate Mean

Step 1: Find the class mark $x_i$ for each class i.e.,

$x=\dfrac{1}{2}(\text{lower limit + upper limit})$

Step 2: Calculate $f_i \cdot x_i$ for each $i$.

Step 3: Apply the formula of Mean $= \dfrac{\sum(f_i \cdot x_i)}{\sum f_i}$


Example: 

(1) Direct method of calculating mean: 4,6,8,10

The mean is calculated by adding all of the numbers and dividing the sum of numbers by the total count of numbers.

Mean $= \dfrac{4+6+8+10}{4} = \dfrac{28}{4} = 7$


(2) Calculate mean of the following numbers

Class Interval

0-10

10-20

20-30

30-40

40-50

Frequency

12

16

6

7

9


Ans:

Class Interval

Frequency $f_i$

Class Mark $x_i$

$(f_i \cdot x_i)$

0-10

12

5

60

10-20

16

15

240

20-30

6

25

150

30-40

7

35

245

40-50

9

45

405


$\sum f_i=50$


$\sum(f_i \cdot x_i)=1100$


Median

The median is the central or middle value of a data set.

$\text{Median}=\dfrac{\text{Number of observations}+1}{2}$

Formula for Calculating Median:

Median $=l+h\left ( \dfrac{\left ( \dfrac{N}{2}-cf\right )}{f} \right )$


Example: 

(1) Direct method of calculating Median 1,2,4,5,7,8,10,12,13,15,16,18,20

Manually, here the median obtained is 10 as it is the middle term from the data set.

Using formula, $\text{Median}=\dfrac{\text{Number of observations}+1}{2}$

$\text{Median}=\dfrac{13+1}{2}=7$

Hence median = 10


(2) Calculate median from the given table below:

Class Interval

0-8

8-16

16-24

24-32

32-40

40-48

Frequency

8

10

16

24

15

7


Ans: 1st calculate the cumulative frequency that is, by adding the sum of each frequency in a frequency distribution table to its predecessors

Class Interval

Frequency

Cumulative Frequency (cf)

0-8

8

8

8-16

10

18

16-24

16

34

24-32

24

58

32-40

15

73

40-48

7

80


$N=\sum f_i=80$


As we have to find the median then, $N = \dfrac{N}{2} = \dfrac{80}{2} = 40$

The cumulative frequency just above 40 is 58, and the associated class is 24-32.

Therefore, l = 24, h = 8, f = 24, cf = c.f. of preceding class = 34, and $\dfrac{N}{2}=40$.

Hence, Median $=l+h\left ( \dfrac{\left ( \dfrac{N}{2}-cf\right )}{f} \right )$

$\Rightarrow  24+8 \dfrac{(40 – 34)}{24} = 26$ 


Mode

The most frequently occurring data item is mode.

Example: (20,20,21,21,21,22,22,23,23,23,23,23,23,24,25,26

In the given data set, the mode is obtained by selecting the most frequently occurring item in the data set.

Hence the mode here is 23.

Note: The relationship between the mean, median, and mode is

Mode = 3(Median) – 2(Mean)


Terms in Statistics and Probability

1. Random Experiment

If an experiment is repeated several times under similar conditions and does not produce the same outcome each time, but the outcome in a trial is one of several possible outcomes, the experiment is referred to as a random event or a probabilistic event.


2. Elementary Event

The outcome of each random event is referred to as an elementary event. When a random event occurs, each associated outcome is referred to as an elementary event.


3. Sample Space

The set of all possible outcomes of a random event is referred to as the Sample Space. For example, when a coin is tossed, there are two possible outcomes: head or tail.

Sample Space = { H,T}

Or

Assume we threw a dice at random, and the sample space for this experiment includes all possible outcomes of throwing a dice.

Sample Space = { 1,2,3,4,5,6}


4. Random Variables

Random variables are variables that represent the possible outcomes of a random experiment. They are classified into two types:

Discrete Random Variables and Continuous Random Variables

Discrete random variables accept only distinct values that can be counted. Continuous random variables, on the other hand, have an infinite number of possible values.


5. Independent Occurrence

When the probability of one event has no effect on the probability of another, the events are said to be independent of one another. For example, if you flip a coin and roll a dice at the same time, the probability of getting a 'head' is independent of the probability of rolling a 6.


6. Expected Value

A random variable's expected value is its mean. For a random experiment, it is the assumed value that is taken into account. It is also known as anticipation, mathematical anticipation, or the first moment. For example, if we roll a six-sided die, the expected value will be the average of all possible outcomes, which is 3.5.


7. Variance

The variance basically tells us how the values of the random variable are distributed around the mean value. It specifies the sample space's distribution across the mean.


Concept of Probability in Statistics

Understanding probability helps us understand statistics and how to apply it because data used in statistical analyses often contains some amount of random variation.

Probability and statistics are in fact closely entwined linked. 

For example, When an anthropologist studies a small group of people from a larger population, the results of his research will include some random variations that he must account for using statistics.


Application of Probability

In real life, probability has a wide range of applications. Some of the common applications we see in our daily lives when checking the outcomes of the following events:

  • Selecting a card from a deck of cards

  • Tossing a coin

  • Tossing a dice into the air

  • Taking a red ball from a bucket full of red and white balls

  • Taking part in a lucky draw

  • Other Important Probability Applications

  • It is used in a variety of industries for risk assessment and modeling.

  • Weather forecasting, also known as weather prediction, is the prediction of changes in the weather.

  • The probability of a team winning a sport based on its players and overall strength.

  • In the share market, chances of getting the hike of share prices.


Application of Statistics

Statistics has numerous applications in mathematics as well as in everyday life. The following are some examples of statistics applications:

  • Applied statistics, theoretical statistics, and mathematical statistics are all types of statistics.

  • Data mining and machine learning

  • In society, statistics

  • Statistical analysis

  • The application of statistics to the mathematics of the arts


Example of Probability and Statistics:

Example 1:  Find the mean, and mode of the following data

2, 4, 7, 5, 10, 7, 12, 6, 4, 3.

Ans: Given data is 2, 4, 7, 5, 10, 7, 12, 6, 4, 3. 

Total elements in the data is = 10

Sum of all the numbers = 2 + 4 + 7 + 5 + 10 + 7 + 12 + 6 + 4 + 3 = 60

$\text{Mean }= \dfrac{\text{Sum of all the numbers}}{\text{Total number of items}}$

$\text{Mean }= \dfrac{60}{10} = 6$

Mode = 7 as it is occurring 2 times.


Example 2: Five blue, four green, and five red balls are stored in a bucket. Manooj is asked to pick two balls at random from the bucket without replacing them, and then to pick one more ball. What is the possibility he picked 2 green balls and 1 blue ball?

Ans: Five blue, four green, and five red balls gives a total count of balls = 14

The probability of drawing 1 green ball from the four $=\dfrac{4}{14}$

As it is asked to pick 2 green balls so the probability of drawing another green ball $= \dfrac{3}{13}$

Probability of drawing 1 blue ball $= \dfrac{5}{12}$

Therefore probability of picking 2 green balls and 1 blue ball $= \dfrac{4}{14} \times \dfrac{3}{13} \times  \dfrac{5}{12} $

$\Rightarrow \dfrac{2}{7}\times \dfrac{3}{13}\times \dfrac{5}{12}$

$\Rightarrow \dfrac{2\times 3\times 5}{7\times 13\times 2\times 6}$

$\Rightarrow \dfrac{5}{182}$


Solved problems of Previous Year Question

1. One coin is thrown 100 times. What is the probability of getting a tail as an odd number?

Ans: Let us consider p = Probability of getting tail $= \dfrac{1}{2}$

q = Probability of getting head $= \dfrac{1}{2}$

As the coin is thrown 100 times, n = 100, and p + q = 1

Let X be an event of getting tail

So, the required probability $= P (X = 1) + P (X = 3) +….. + P (X = 99)$

$\Rightarrow {}^{100} {C}_{1}\left(\dfrac{1}{3}\right)\left(\dfrac{1}{2}\right)^{99}+{ }^{100} {C}_{3}\left(\dfrac{1}{2}\right)^{100}+\cdots+{ }^{100} {C}_{99}\left(\dfrac{1}{2}\right)^{100}$

$\Rightarrow \dfrac{{ }^{100} {C}_{1}+{ }^{100} {C}_{3}+\cdots+{ }^{100} {C}_{99}}{2^{100}}$

$\Rightarrow \dfrac{\dfrac{2^{100}}{2}}{2^{100}}$

$\Rightarrow \dfrac{2^{99}}{2^{100}} = \dfrac{1}{2}$


2. If a variable takes the discrete values $\alpha − 4, \alpha − \dfrac{7}{2}, \alpha − \dfrac{5}{2}, \alpha − 3, \alpha − 2, \alpha + \dfrac{1}{2}, \alpha − \dfrac{1}{2}, \alpha + 5$ where $(\alpha > 0)$, then the median is ____________.

Ans: Given values are $\alpha − 4, \alpha − \dfrac{7}{2}, \alpha − \dfrac{5}{2}, \alpha − 3, \alpha − 2, \alpha + \dfrac{1}{2}, \alpha − \dfrac{1}{2}, \alpha + 5$

On rearrange the data in ascending order, we get $\alpha − \dfrac{7}{2}, \alpha − 3, \alpha − \dfrac{5}{2}, \alpha − 2, \alpha − \dfrac{1}{2}, \alpha + \dfrac{1}{2}, \alpha − 4, \alpha + 5$

Median $= \dfrac{1}{2} \left[\text{value of 4th item }+\text{ value of 5th item}\right]$

Median $= \dfrac{(\alpha − 2)+\dfrac{\alpha − 1}{2}}{2}$

$\Rightarrow \dfrac{2\alpha − \dfrac{5}{2}}{2}$

$\Rightarrow \alpha − \dfrac{5}{4}$


3. If A and B are two independent events such that $P (A \cap B^\prime) = \dfrac{3}{25}$ and $P(A^\prime \cap B) = \dfrac{8}{25}$, then $P(A) =$ _______.

Ans: Since the two events are independent.

Then, $P (A \cap B^\prime) =  P (A) \times P (B^\prime) = \dfrac{3}{25}$

$\Rightarrow P (A) \times (1 − 2 P (B)) = \dfrac{3}{25}$ ….(i)

Similarly, $P (B) \times (1 − P (A)) = \dfrac{8}{25}$ ….(ii)

Solving equation (i) and (ii), we get

$P (A) = \dfrac{1}{5} \text{ and } \dfrac{3}{5}$


Practice Problems

1. Find the mean, median, and mode of the following data.

6,10,8,10,10,7,11,12,15

Ans: Mean=10

Median=10

Mode = 10


2. If one card is drawn at random from a deck of 52 cards, what are the chances that it will be a king or a spade?

Ans: $\dfrac{4}{13}$

Hint: Divide king and spade with total deck of cards and calculate the probability of king and spade by, P (King or spade) = P (king) + P (spade)− P (King and spade)


Conclusion

Probability and statistics is a scoring topic in mathematics, with questions that are very basic and simple. Probability and statistics has an overall weightage of 11 to 15 marks, with main questions coming from topics in statistics such as Conditional Probability, Properties of Conditional Probability, Multiplication Theorem, Bayes Theorem, and Probability Distributions, Theorem of Total Probability, Descriptive Statistics, Inferential Statistics and so on. Try to practise as many questions as possible. Download the probability and statistics notes pdf now.

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FAQ

1. What are the basic rules of probability?

If A and B are the two events:

Addition Rule: $P(A\cup B) = P(A) + P(B) – P(A\cap B)$

Multiplication Rule: $P(A\cap B) = P(B) \cdot P(A|B)$

2. What role does statistics play in everyday life?

Statistics encourages you to use legitimate methods for gathering data, conducting appropriate tests, and effectively presenting the results. Measurement is a crucial step in the process of making scientific disclosures, making informed decisions, and making forecasts.

3. Give an explanation of the binomial distribution.

A binomial distribution describes the chances of an event succeeding or failing. In a test, for example, there is a chance of passing or failing.

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