Probability And Statistics Symbols

Introduction

Probability is simply a useful description (in the form of a mathematical model) for experiments whose exact outcome is difficult to predict in advance. When you toss a coin, it's tough to know in advance if a head or a tail will appear. When you can't predict the exact outcome, it's often useful to try to characterize every outcome that could occur along with a numerical description as to which are the most likely to occur. The numerical description you choose can be based on your experience, knowledge of physics, what makes for easy calculations, or many other factors. Statistics deal with a variety of data. You may be interested in finding the most popular or most used items in your dataset.

In such cases, we may edit the record to provide analysis and conclusions.

This concept of dealing with data analysis, interpretation, and display of data in a more meaningful way is statistics. In this tutorial, we will discuss probability and statistics symbols.

Probability: Definition

Probability is the possibility of happening an event

$$\mathrm{Probability(event)\:=\:\frac{Favourable\:outcomes}{Total\:outcomes}}$$

Probability is the percentage of success. Probability is used to describe an outcome function for fixed parameter values. For example, if you toss a coin 10 times and it is a fair coin, what is the probability that it comes up heads each time? This can be calculated by the above formula.

Properties of Probability

• The range of probability is from 0 to 1

• The probability will be one for a sure event.

• The probability will be zero(0) for an impossible event.

• The total probability of events is 1.

Statistics

Statistics is a field of mathematics related to research that collects, analyzes, interprets, presents, and organizes data in specific ways. Statistics are defined as the process of collecting data, classifying it, presenting it for easy interpretation, and then analyzing it. Statistics are also referred to as concluded sample data collected through surveys or experiments.

In mathematical statistics, there are two widely used methods of data analysis.

Descriptive Statistics.

It is used to describe the data collected and summarize the data and its characteristics along with central tendency and variance measurements.

Inference Statistics

This statistical technique is used to conclude the data. Inference statistics require a statistical test to be performed on the sample and draw conclusions by identifying the differences between the two groups.

Symbols Used in Probability and Statistics

Statistics deals with a variety of data. You may be interested in finding the most popular or most used items in your dataset. In such cases, we may edit the record to provide analysis and conclusions. This concept of dealing with data analysis, interpretation, and display of data in a more meaningful way than statistics.

List of probabilities and statistical symbols

• $\mathrm{P(A\:\cap\:B)\:=\:probability\:that\:of\:events\:A\:and\:B}$

• $\mathrm{P(A)\:=\:probobility\:of\:event\:A}$

• $\mathrm{P(A\:|\:B)\:=\:probability\:that\:event\:A\:given\:event\:B\:occured}$

• $\mathrm{P(A\:\cup\:B)\:=\:probobility\:that\:of\:events\:A\:or\:b}$

• $\mathrm{std\:(X)\:=\:S.D\:=\:standard\:deviation\:of\:random\:variable\:X}$

• $\mathrm{\sigma^{2}\:=\:variance\:of\:population\:values}$

• $\mathrm{\sigma_{x}\:=\:standard\:deviation\:of\:random\:variable\:X}$

Probability Distribution Table vs Frequency Distribution Table

Probability Distribution

It provides the possible results for any random event. It is also defined as a set of possible results for randomized experiments based on the underlying sampling space. These settings can be a set of real numbers, a set of vectors, or any set of entities. It's part of probability and statistics.

Frequency Distribution

It shows a summarized group of data classified into mutually exclusive classes and the number of occurrences within the class. This is a way to display disjointed data, especially to show election results, the income of people in a particular region, etc.

Expectancy in Probability vs Mean in Statistics

Expectancy in probability:

It is equal to the sum of the products of each possible result and its probabilities and is expressed as an expected value expression. If the probabilities of each result can occur in the same way, the expected value is the arithmetic mean of all the results. Probability and statistics use the expected formula to find the mean of the random variable X represented by E (x). This is also known as the average, mean, or first moment.

Mean in statics:

In statistics, the average (mean) of a particular set of observations will be equal to the sum of all the given observations in the data collection divided by the total number of given observations in the data. However, general procedures and formulas depend on the type of data specified, grouped data, or ungrouped data.

Variance and Standard Deviation in Probability vs Statistics

In probability theory and statistics, the variance is the expected value of the square of the deviation from the population or sample mean of the random variables. Variance is a measure of spread. In other words, it is a measure of how far a set of numbers is from the average.

Variance is a statistical measure used to determine the spread of a number in a dataset over a mean or average. The square of the standard deviation indicates the variance. Use the variance to evaluate how stretched or compressed the distribution is.

There are two types of variances in statistics: sample variance and population variance. The symbol of variance is given by 𝜎2.

$$\mathrm{\sigma^{2}\:=\:\frac{\Sigma\:(x_{i}\:-\:x)^{2}}{n\:-\:1}\:Where\:x_{i}\:=\:value\:of\:one\:observation}$$

$\mathrm{\overline{x}}$= average of all observations

n = number of observations

In statistics and probabilities, the standard deviation of a given variable is the mean distance of the given variable from the mean. This shows how the random variables are distributed near the mean. A small standard deviation indicates that the variables are distributed close to the mean.

$$\mathrm{Standard\:deviation\:=\:\sqrt{\frac{\Sigma(x_{i}\:-\:\underline{x})^{2}}{n\:-\:1}}}$$

Solved Examples

1)Find the mean of the following numbers. $\mathrm{1\:,\:1\:+\:x\:,\:1\:+\:2x\:,\:+\:...........\:+\:1\:+\:100x}$.

Answer − Given numbers are, $\mathrm{1\:,\:1\:+\:x\:,\:1\:+\:2x\:,\:+\:...........\:+\:1\:+\:100x}$

Total number of terms, n= 101

$$\mathrm{Mean(\overline{x})\:=\:\frac{1\:,\:1\:+\:x\:,\:1\:+\:2x\:,\:+\:...........\:+\:1\:+\:100x}{101}}$$

$$\mathrm{=\:\frac{1}{101}\times\:\frac{101}{2}\:[1\:+\:(1\:+\:100x)]}$$

$$\mathrm{=\:1\:+\:50x}$$

2) Find the relation between variance and standard deviation.

Answer − We know that the standard deviation $\mathrm{=\:\sqrt{\frac{\Sigma\:(x_{i}\:-\:\overline{x})^{2}}{n\:-\:1}}}$ and variance $\mathrm{=\:\frac{\Sigma\:(x_{i}\:-\:\overline{x})^{2}}{n\:-\:1}}$

On squaring standard deviation we get variance hence,

$$\mathrm{standard\:deviation\:=\:\sqrt{variance}}$$

3)What will be the probability of getting 5 on a fair die?

Answer − Let A denote the event “getting 5.”

According to the question, sample space will be $\mathrm{\lbrace\:1\:,\:2\:,\:3\:,\:4\:,\:5\:,\:6\:\rbrace}$

$$\mathrm{P(A)\:=\:\frac{1}{6}}$$

4) What is the probability of getting 2 on a fair die, given you’ve rolled an even number?

Answer − You’ll note that the sample space has been now reduced to S′={2,4,6}. Assuming B to be the event “rolling an even number,” the probability is now −

$$\mathrm{P(A\:given\:B)\:=\:\frac{1}{3}}$$

5) Find the standard deviation if the variance is 36.

Answer − We know that the $$\mathrm{\mathrm{standard\:deviation\:=\:\sqrt{variance}}}$$

$$\mathrm{standard\:deviation\:=\:\sqrt{36}}$$

$$\mathrm{standard\:deviation\:=\:6}$$

Conclusion

Probability is the possibility of happening an event. Probability is used to describe an outcome function for fixed parameter values. Statistics are defined as the process of collecting data, classifying it, presenting it for easy interpretation, and then analyzing it.

FAQs

1. What is probability?

Probability is the possibility of happening an event.

2. What is a variance?

Variance is a statistical measure used to determine the spread of a number in a dataset over a mean or average. The square of the standard deviation indicates the variance.

3. What is the use of Statistics?

Statistics is a study of data that is how to collect, summarize, and present it.

4. What is Probability Distribution?

In statistics, the probability distribution indicates the likelihood of each outcome of a random experiment or event.

5. What is Sample Space?

It is defined as the set of all possible outcomes. In other words, The sampling space can be called a collection of all possible results in a random experiment.