## Probability

The probability theory is used to study random events in a logically valid manner as it provides a mathematical framework for doing so. However, the probability of an event occurring is a numerical value that indicates how probable it is that the event will occur.

This number is always in the range of 0 to 1, with 0 indicating impossibility and 1 indicating certainty in the situation. A fair coin toss is a typical example of a probability experiment, in which the two potential outcomes are either heads or tails, and the outcome is determined by chance.

In this situation, there is a 50 percent chance of having either a head or a tail. However, in a sequence of coin flips, we may obtain more or less than 50 percent heads, depending on the circumstances. In the long term, however, the frequency of heads will inevitably grow as the number of flips increases, and eventually reach 50 percent.

Thus the likelihood of an event occurring let’s suppose A is the number of ways in which A may occur divided by the total number of other possibilities (number of all other possibilities).

As a result, we may refer to this as the “counting definition of probability,” mostly because each conceivable event to count is frequent and discrete, which allows us to simplify the concept of probability even more. However, it is still beneficial to study the underlying rules in this case. Online probability calculator help us how find the probability.

## Summation

When dealing with a big number of data points needed to be added or summed up, the process of addition is done using the summation. Summation notation is an extremely helpful and shorthand way when attempting to write an extremely big number and summation notation calculator explains how to find summation value.

For instance, the summation in the sequence [11,9,7,5..] is defined as the sum of the values of each of the numbers in the series.

To put it another way, summation notation aids in the representation of a concise form for the addition of a large number of data points. To represent summation, we use the sign **Σ**, which is a Greek upper letter known as sigma.

Whenever a sequence is required to add from left to right, it might result in a partial sum, running total, or prefix sum, which would be the result of running intermediate results.

The following is the format in which the summation notation is generally represented as given below. Read the following explanations of what each symbol in the summation formula represents:

i = nmai = an an+an+1+an+2+….+am−2+am−1+am

**Σ **= Summation Sign

**ai** = Typical element

**m **= Upper limit of summation

**n** = Lower limit of summation

**i **= Index of summation

Meanwhile the expression mentioned above is read as the sum of **a** sub** i** equals **n** to **m**.

## Expected Value

The expected value as in probability theory is a theoretical value that indicates the average return that would be obtained if the activity were performed indefinitely. To calculate the expected value the weighted average of all potential outcome values is calculated whereby the probability of the provided results is given as the weight.

Expected value can be calculated in the same way using the formula in which the results of the event is denoted using the **x** whereas the probability of the occurring event is represented as **P(x)**

**E(x) = x**1** * P(x**1**) + x**2** * P(x**2**) + x**3** * P(x**3**)…**

In the equation above we can include as many **x****z**** * P(x****z****)** as we want, since there are possible outcomes for the action we are calculating or in other words which is the maximum number conceivable.

Moreover, for the expected value formula, there is also a shorter way to express and calculate as

**E(x) = ****Σ****x * P (x)**

It should be noted that the formula comprising the sigma depicts the same thing as mentioned in the summation explanation. The sigma or summation represents the weighted mean of the potential outcomes, wherein the probability of each event taking place in the event is represented as the weight.