# A Modern Introduction To Probability And Statistics With Solved Examples

Probability and statistics are useful guides for studying numbers. Probability lets us determine how likely events are to occur, such as predicting whether it will rain. Statistics, on the other hand, is concerned with the collection and comprehension of data, similar to looking at numbers to discover important information. They work together to help us make informed judgements and see patterns in the information we encounter. This article discusses various ideas in probability and statistics.

## What is Probability And Statistics?

Probability is concerned with the likelihood of events occurring, assigning a value between 0 and 1 to quantify uncertainty. It assists us in understanding the likelihood of various outcomes in uncertain situations, such as forecasting weather or game results. Statistics, on the other hand, is the process of gathering, analysing, and interpreting data in order to reach meaningful conclusions. It provides methods for summarising information, identifying patterns, and making educated judgements in a variety of domains, hence improving our understanding of uncertainty and unpredictability in real-world circumstances.

### Probability Definition

Probability refers to the likelihood of the outcome of any random event. This word refers to determining the likelihood that an event will occur. For example, what is the chance of receiving a head if we flip a coin in the air? The answer to this question depends on the number of possible outcomes. In this case, the conclusion might be either head or tail. As a result, the probability of receiving a head is half.

Probability is a measure of the possibility that an event will occur. It evaluates the event’s certainty. The probability formula is provided as follows:

P(E) = Number of Favorable Outcomes/Number of total outcomes

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

Here,

n(E) = Number of event favourable to the event E

n(S) = Total number of outcomes

### Statistics Definition

Statistics is the study of how data is collected, analysed, interpreted, presented, and organised. It is a method for gathering and summarising data. This has numerous applications, ranging from tiny to large-scale. Stats are utilised in various types of data analysis, including the examination of a country’s population and economy.

Statistics has a broad use in a variety of subjects, including sociology, psychology, geology, and weather forecasting. The data acquired for analysis may be quantitative or qualitative. Quantitative data are classified into two types: discrete and continuous. Continuous data has a range, but discrete data has a set value. There are numerous terms and formulas utilised in this topic.

Also Read: Properties of Triangle: A Complete Guide

Random Experiment: An experiment is a series of steps that produce obvious findings. A random experiment is one in which you cannot predict the exact outcome.

Outcome: An outcome is any potential result in a set of results known as a sample space, denoted by the symbol S. For instance, when flipping a fair coin, the sample space is {heads, tails}.

Sample Space: The set of all possible results of an experiment. Similar to a coin flip, the sample space is {heads, tails}.

Event: An event is any portion of a sample space. If an event A occurs, it indicates that one of the A outcomes has occurred. For example, rolling an even number on a fair six-sided die and obtaining 2, 4, or 6 indicates that event A occurred. If you get one, three, or five, event A did not occur.

Trial: A trial is any time you do an experiment, such as flipping a coin. In the coin-flipping experiment, each flip is treated as a trial.

Mean: The mean of a random variable is the average of all possible values during a random experiment.

Expected Value: The expected value is the average of a random variable. For example, if we roll a six-sided die, the expected value is 3.5, which represents the average of all possible outcomes.

## Probability And Statistics Formulas

For two events A and B, the probability formulas are listed below:

Statistics Formula are listed below:

## Solved Examples

1. Find the mean and mode of the following data: 2, 3, 5, 6, 10, 6, 12, 6, 3, 4.

Solution,

Total Count: 10

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

Mean = (sum of all the numbers) / (Total number of items)

Mean = 60/10 = 6

Again, Number 6 is occurring for 3 times, therefore Mode = 6

1. What is the chance that Raj will choose a marble at random that is not black if the bowl contains three red, two black, and five green marbles?

Solution,

Total number of marble = 10

Red and Green marbles = 8

So P (not black) = (number of red or green marbles) / (total number of marbles)

= 8 /10

= 4/5

1. Find the median and mode of the following marks (out of 10) obtained by 20 students: 4, 6, 5, 9, 3, 2, 7, 7, 6, 5, 4, 9, 10, 10, 3, 4, 7, 6, 9, 9

Solution,

Given,

4, 6, 5, 9, 3, 2, 7, 7, 6, 5, 4, 9, 10, 10, 3, 4, 7, 6, 9, 9

Ascending order: 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 9, 9, 9, 9, 10, 10

Number of observations = n = 20

Median = (10th  + 11th observation) / 2

= (6 + 6) / 2

Median = 6

Most frequent observations = 9

Hence, the mode is 9.

1. In a deck of cards, what is the probability of drawing a red card?

Solution,

Total number of cards in a deck = 52

Total number od Red cards in a deck = 26 (hearts + diamonds)

P (Red Card) = 52 / 26

⇒ P (Red Card) = 2 / 4

⇒ P (Red Card) = 1 / 2 or 0.5 or 50%

## FAQs

What is the definition of independent events in probability?

Independent events are ones in which the outcome of one event does not influence the outcome of another. For example, flipping a fair coin is a distinct occurrence.

What is probability, and why is it relevant in everyday life?

Probability refers to the possibility of an event occurring. It is necessary in everyday life to make gained judgements based on the probability of various outcomes.

How does the mean differ from the median and mode in statistics?

The mean is the average, the median is the midpoint, and the mode is the most common value in a data set.

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