Probability
It is always a good practice to foresee or partially determine the final output before stepping into any venture of real life. Whether it is business enterprise decisions, cricket championships, or the weather forecast, prediction models always help practitioners in estimating their outcome. The prediction model is the subsidiary area of Probability, a branch of mathematics that deals with the possibility or expectedness of an event. Formally, the Probability of an event can be determined by a ratio, which discusses the total number of outcomes and the possible chances of the occurrence of a particular event, i.e.,
P(A) = No. of desired outcomes
Total possible outcomes
= n(E)/n(S)
Let discuss and simplify the enigma of Probability by an example. For instance, we have to determine the Probability of obtaining an odd number after rolling dice for one single time. In solution will first decide the Sample Space, which will include all possible outcomes, in our case, it will be
S = {1, 2, 3, 4, 5, 6}
And as it has six occurrences, so the n(S) is 6. Our favorable outcomes are the odd numbers hence set of desired outcomes will contain {1, 3, 5}, and consequently,
n(E) = 3
Putting them in probability formula of P(A) = n(E) / n(S), we will have
p(A) = 3 / 6 = ½ = 0.5
Always remember that the probability of an event must only be between 0 and 1.
Let consider a case in which A & B are two mutually exclusive events then,
P (A ∩ B) = 0
and,
P (A U B) = P (A) + P (B).
Clarify these expressions with this example, in which event A will carry the numbers greater than or equal to 4 in a dice roll, i.e., {4, 5, 6}. Whereas the event B will have the numbers lesser than 4 in a dice roll, i.e., {1, 2, 3}. So now,
P(A ∩ B) = 0
P(A U B) = P (A) + P (B)
P(A U B) = {1, 2, 3, 4, 5, 6}
Expected Value
The concept of expected value is similar to the idea of anticipating one most possible case out of the entire set of different scenarios. Generally, it has been unanimously believed that the human conscience always expects an output or a reaction after executing any successful action. In terms of applied philosophy, there is certainly an underlying cause behind any effect and vice versa.
As we dive deep inside, we will come across the theoretical aspects of the expected value. Its mathematical formula is pretty straightforward, which deals with the product of two quantities.
E.V = P(x) * n
Where P(x) represents the probability of a specific event, and n is the number of times that event would happen. The formula mentioned above can be changed under two significant circumstances, i.e., Binomial random variable, or variable for Multiple events.
Let narrow down both cases separately. In the case of the Binomial random variable, you will only be left with two options to expect as the resultant value. For example, if you need to determine the probability of having a tail side of the coin after ten trials of toss. Then you should find the simple probability of getting tail in a single trial, i.e., 1/2 or 0.5. Hence, after multiplying it with a number of trials, you will end up with the final expected value of having a tail side of coin after ten times of toss.
E.V = P(x) * n
= 0.5 * 10 = 5
Now, let's examine the case for variables of multiple events. Unlike binomial, this case deals with the events that will have more than two probabilities.
E.V = P(X) * ∑n
In this formula, the probability of a single event will be multiplied with the sum of all outputs of events.
Math is a conceptual subject and one cannot survive if he/she wants to memorize the concepts instead of learning them. In order to learn concepts of various math functions, one has to practice a lot to learn with effectiveness.
There are 2 ways to practice math in case someone wants to learn various different concepts. The one is the manual way and the other is the digital way. The manual ways to calculate are defined above. The other way of practice is the digital way in which calculators are used instead of manual formulas. There are easily available online calculators like Probability Calculator & Expected Value Calculator which anyone can use to master their subject i.e math.