The Probability

Probability is a measure or estimation of the likeliness or likelihood that an event will occur. Probability is used to quantify an attitude of mind towards some proposition of whose truth we are not certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty), we call probability. Thus the higher the probability of an event, the more certain we are that the event will occur. A simple example would be the toss of a fair coin. Since the the 2 outcomes are deemed equiprobable, the probability of "heads" equals the probability of "tails" and each probability is 1/2 or equivalently a 50% chance of either "heads" or "tails".                              Source - Wikipedia

Probability Questions: -   Let us find the prob for each event when a coin is tossed five hundred times and the frequency of the two events is given as Head: Three Hundred, tails = two hundreds.

Solution : - We are given that the total number of trials is five hundred. Therefore, the number of times E happens, i.e., the number of time the head comes up is three hundred.  Therefore the probability of head = the number of head / Total number of the trials Or P (E) = 300/ 500 = 0.6. Similarly the probability of tail = the number of tail / Total number of the trials Or P (F) = 200/ 500 = 0.4. Hence the prob of success is zero point six and prob of failure is zero point four. The point which is to be noted down is that the total sum of both the probability is equal to one.

Example: - Let us find the prob, when the two coins are tossed simultaneously for four hundred and eighty times and we get the following events. Two heads: one hundred and twenty times (120), One head:  Two hundred and three times. (203), No head:  one hundred and fifty seven time (157)

Solution:-  (1) The probability of getting two heads = the total number of heads divided by the total number of chances. The probability of getting two heads = 120 / 480 = ¼ = 0.25. The probability of getting one heads = 203 / 480   = 0.42292

The probability of getting no heads = 157 / 480 = 0.32708

We can see that P (E₁) + P (E₂) + P (E₃) = 1, cover all the outcomes of a trial.
Or P (E₁) + P (E₂) + P (E₃) = 0. 25 + 0.42292 + 0. 32708 = 1

Example:- The record of a weather station  shows that the out of past two hundred consecutive days, its weather forecast is correct for one hundred and forty five ( 145)  times. Let us calculate the probability that on a given day it was correct and the probability that on a given day it was not correct.

Solution: - The record of a weather station is available for two hundred consecutive days. The Probability P (E₁) that on a given day the forecast was correct = the number of days when the forecast was correct/ Total number of days for which record is available  Or P (E₁) = 145 / 200 = 0.725. P (E₂) that on a given day it was not correct = 55/200 = 0.275, Notice that P (the forecast was correct + the forecast was not correct) = 0.275 + 0.725 = 1

How do you Find Probability: -The word probably, doubts most probably, chances etc are used to define the uncertainty. The uncertainty of the probably etc can be measured numerically by means of probability. Therefore the probability started with gambling, it has been used extensively in the field of physical sciences, commerce, etc. Let there are n number of trials. The probability of an event E happening, is given by P (E) =
Number of trials in which the event happened / the total number of trials.

Let there are n number of trials. The probability of an event E happening, is given by P (E) = Number of trials in which the event happened / the total number of trials.

Derivative Of Tanx?

Derivative of tan x is a tool which is used to find the differentiation of tan x. Before we understand the differentiation of tan x it is important to understand about tan x. If we will describe tan x in terms of sin x and cos x, then it can be defined as the ratio of Sin x and Cos x.

Mathematically, tan x = Sin x/ Cos x

It is important to note that tan x is the reciprocal of the Cot x and vice versa. Apart from that it is also important to note that the derivative is simple denoted as d/dx. Now the question arises what is the Derivative of Tanx? So it can be simply written as d/dx (tan x).

The value of d (tan x)/dx is equal to Sec^2 x. Now in the coming context, you will learn how the derivative of tan x that is d (tan x)/dx is equal to Sec^2 x. This can be explained by the basic concepts of trigonometry and rules of derivative. 

So to understand the derivative of tan x it is really important to have thorough knowledge of the trigonometry as well as the differentiation. The proof of the derivative of tan x can be done with the help of the rules of the differentiation and trigonometry which are as shown below:-
 
We know that tanx = Sin x/ Cosx
 
So the derivative of tan x, that is, d (tan x)/dx = d (Sin x/Cosx)/ dx
 
We know that by the divisibility rule of differentiation,
 
That is, d (u/v)/ d x = (v du/dx – u d v/dx)/ v^2
 
So d (tan x)/dx = d (Sin x/Cos x)/ dx
= (Cos x d(Sinx)/dx- Sin x d(Cosx)/dx )/ Cos ^2 x
= (Cos x. Cos x – Sin (- Sinx) ) / Cos^2 x
= ( Cos^2x + Sin^2x) / Cos ^ 2x
 
We know that Cos^2x + Sin^2x = 1
 
Therefore, d (tanx)/dx = 1/ Cos^2 x
 
We Know that 1/ Cos x = Sec x (This is as per the trigonometric rules)
 
So d (tan x)/dx = 1/ Cos^2 x = Sec^2 x (This is in line with the above equation)
 
Hence we have seen the proof that d/dx of tanx is equal to Sec^2 x, that is, d (tanx)/dx = 1/ Cos^2 x = Sec^2 x. But from above proof, we have learnt that without knowing the differentiation rules as well as trigonometric rules, it is impossible to prove the differentiation result of the trigonometric function.

Differentiation of one function can also help in finding the differentiation of other functions. So it is very important that we should know the differentiation of each and every function. For example sometimes the question requirement is to find the differentiation of the trigonometric as well as exponential, so in that case to find the differentiation we need to know the rules along with the individual derivative of the functions.
 
We can make use of Second Derivative Calculator to find out further derivatives.