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Probability is a word which is used in our everyday life. Many synonyms are used in place of the word ‘probability’ like chance, luck, fortune, likelihood, odds, etc. We use these words vaguely, but we generally don’t try to quantify them i.e. measure the chances of something to happen. This is where Probability as a subject is born and is utilized. Probability was first conceived to study games like playing cards, rolling of dice and tossing of coins. Slowly but steadily the subject evolved to measure the chances of not just games, but spread its span to many other fields like Physics, Finance, Economics, banking and insurance and Social sciences like Psychology. All these fields use probability and statistics in some way or the other to record data and to measure the chances of happening and non-happening of an event. For Ex: a banker uses probability and statistics to assess a customer’s loan-repaying capacity. Basically, he is dong a risk assessment. So, it’s needed for one to know how probability and statistics work.

Expected value is the synonym for mean of a random variable. It is the measure of central location for the random variable. It is the weighted average of the values that the random variable may assume. Though expected value gives the mean of the random variable, we need to measure variance in some cases. Variance is the weighted average of the squared deviations of a random variable from its mean. The weights are the probabilities. While computing the variance of a random variable, the deviations are squared and then weighted by the corresponding value of the probability function. Standard Deviation is used to indicate how much the individuals in a group differ. If individual observations in a group differ, the standard deviation is big and if individual observations do not differ from group, then the standard deviation is small. Note that standard deviation of a sample and population differ and they are calculated differently. 

The expectation value of a function f(x)  in a variable x is denoted f(x) or E(f(x)).  For a single discrete variable, it is defined by


where µ is the mean for the variable. 

The standard deviation of a population is measured by the following formula

                          σ = √[Σ(xi- x)2÷ N] 

 Where σ is the population standard deviation, x is the population mean, xi is the ith element from the population, N is the number of individuals in the population.
The standard deviation of a sample is measured by the following formula

                        S =   √[Σ(xi- x)2 ÷ (n-1)]

Where s is the sample standard deviation, x is the sample mean, xi is the ith element from the population, n is the number of individuals in the sample.
And Standard Deviation is equal to the square root of the variance. So, variance is equal to the square of standard deviation.  

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