Probability & Statistics: Concepts, Notes, Theorems & Solved Examples

Probability and Statistics are two main concepts used in mathematics to solve various problems. Probability is used in those concepts to compute the chances of getting or not getting a result. Statistics is a field in which various types of data is handled using various types of techniques. Probability and statistics are topics that are taught from classes 9 to 12 of varying difficulty. These two chapters are also very important for Class 10 and Class 12 Board Examinations. Let's study these topics in detail.

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What is Probability?

Probability is known as the possibility of getting the outcomes related to any specified event. We can check the possibility of any event and how many times the event can happen. For Example, if we toss a coin 1 time, there can be a head or tail. Now, we can find the probability of getting a head or tail. We can find that the probability of getting a head or tail is equal. It means that the probability of getting a head or tail is ½. We can find the certainty of any event with the help of the concept of probability.

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Formula of Probability

Formula of Probability is given by,

Probability of an event = Number of Favourable Outcomes / Number of Total Outcomes

or

P(E) = n(E) / n(S)

Where,

• P(E) = Probability of an event
• n(E) = Number of events which are favourable to the event E
• n(S) = Number of total outcomes which can occur

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Important Formulas for Probability

All important formulas of Probability are as follows:

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Types of Probability

There are 3 types of Probability which are as follows:

Theoretical probability

Formula of Theoretical probability is:

P(E) = Number of Favourable Outcomes / Number of Total Outcomes

For Example, if we toss a coin, we can get a head or a tail;

P(getting a head) = ½

P(getting a tail) = ½

Experimental Probability

Formula of Experimental probability is:

P(E) = Number of Favourable Outcomes / Total number of times experiment is done

For Example, if we toss a coin 5 times, we get head 2 times and tail 3 times;

P(getting a head) = 2/5

P(getting a tail) = 3/5 

Axiomatic Probability

In Axiomatic probability, there are 3 axioms which are applicable to all types.

These 3 axioms are:

1. Probability is more than or equal to zero for any event.
2. Set of all possible outcomes which can occur in an event are defined as Sample Space.
3. Probability of A or Probability of B is addition of probability of A and the probability of B.

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Rules of Probability

Rules of Probability are as follows:

1. Probability of an impossible event is null.
2. Maximum Probability of any event can be its total number of possible outcomes.
3. Probability exists between 0 and 1. Probability can be zero also.
4. Probability cannot be negative.
5. Probability of A or Probability of B is addition of probability of A and the probability of B.

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What is Statistics?

Study of collection, presentation, organisation and analysis of data is known as Statistics. We can collect and summarise data with the help of Statistics. We can use this concept in various small and big industries. We can study the population of a country, sociology, weather forecasting, geology, and many more. Statistics collect various types of data whether it is qualitative or quantitative.

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Important Formulas of Statistics

All important formulas of Statistics are as follows:

Here, x means the given item, and n means the total number of items.

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Measures of Central Value

There are 3 measures of Central Value:

Mean

Average of a given set of numbers is known as the mean.

For Example: 5, 7, 9, 11

Mean = 5+7+9+11 / 4

Mean = 8

Median

Middle Value in a given set of data is known as median.

For Example: 4, 6, 8, 10, 12

Here Median is 8.

Mode

Most Occurring item in a given set of data is known as mode.

For Example: 1, 2, 2, 3, 7

Mode = 2

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Measures of Spread

There are mainly 5 measures of Spread:

Range

Difference of maximum and minimum value in a given set of data is known as Range.

For Example: 2, 4, 6, 8, 10

Range = 10-2 = 8

Quartile

Arranging the data in quarters is known as Quartile.

For Example: 1, 2, 3, 4, 5, 6, 7

There are 7 terms in this data.

Q1 = (7+1 / 4)*1 = 2

Q2 = (7+1 / 4)*2 = 4

Q3 = (7+1 / 4)*3 = 6

Here, Q1 is Lower Quartile, Q2 is middle Quartile, and Q3 is Upper Quartile.

Interquartile

Difference of Upper Quartile and Lower Quartile is known as Interquartile.

Fore Example, Q1 = 2 and Q2 = 6

Interquartile = Q2 - Q1 = 6-2 = 4

Variance

A measure which gives an approximate idea about data is known as variance. It is used to calculate standard deviation of a data set.

σ² = ∑ni=1(xi-\(\mu\))² /n

Standard Deviation

We can find how far a value is from the mean by the help of Standard Deviation. It can be found by taking the square root of Variance.

σ = √∑ni=1(xi-\(\mu\))² /n

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Measures of Comparing Data

There are mainly 5 measures of Comparing Data:

Univariate Data

It compares only one type of data. For Example, we can compare the weights of different students in a class.

Weights of students: 55 KG, 57KG, 52KG, 49KG, 60KG

Bivariate Data

It compares two types of data. For Example, Number of trousers sold in a week.

Scatter Plots

We can know how one data is related to another data with the help of scatter plots. For Example: A graph represents the weight and age.

Outlier

Values that are out of range of a given data set are known as Outliers. They are the minimum and maximum extreme values.

For Example: 1, 50, 60, 70, 80, 90, 650

In this data set, 1 and 650 are outliers.

Correlation

Correlations are between those two data sets which are decreased or increased together. It is positive if two data sets increase together.

For Example: As we walk for more time, the distance to cover decreases.

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Other terms used in Probability & Statistics

There are various terms used in Probability and Statistics:

• Random Experiment: Experiment whose result cannot be predicted is known as random experiment.
• Sample Space: Set of all possible outcomes of any experiment is known as Sample Space.
• Random Variables: Variables denoting the possible outcomes are called random variables.

• Independent Event: Events are independent if occurrence of one outcome does not affect the occurrence of another outcome.

• Expected Value: Mean of random variable is known as Expected Value.

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List of Probability Topics

Following are the list of articles related to the topic of Probability

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List of Statistics Topics

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Solved Examples

Ques: Keshav rolled a dice 1 time. What is the probability of getting a number 5. (2 Marks)

Ques: In a bucket, there are 5 blue balls, 4 green balls, and 5 red balls. Sameer picks 2 balls randomly from the bucket without replacement and then one more ball is to be picked. Find the probability that he picked 2 green balls and 1 blue ball? (3 Marks)

Ques: Determine the mean of the following given info set.
10, 20, 36, 12, 35, 40, 36, 30, 36, 40 (3 Marks)

Ques: The marks scored by 5 students in a weekly class test are 7, 9, 6, 4, 2 out of 10. Determine the mean marks of the class? (2 Marks)

Ques: The arithmetic mean is a measure of central tendency and is commonly referred to as the mean. The arithmetic mean is calculated by dividing the total of all the values in a series by the number of items in that series. Arithmetic mean is normally denoted by X¯X¯ which is red as 'XX bar.' It can be computed for unclassified or ungrouped data, as well as discrete or continuous series, as well as classified or grouped data. Calculate the arithmetic mean from the following data. (4 Marks)

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