The Naive Bayes classifier is based on the Bayes theorem. Bayes theorem was given by Rev. Thomas Bayes and is based on the application of conditional probability. Conditional probability for any event e is the probability that e will occur, given that another event has already occurred. Thus, based on this postulate, we can calculate the probability of an event using its prior knowledge.

Below is the formula for calculating conditional probability.

P (H|E) = (P (E|H) * P (H)) / P (E)

Where,

P (H) is the probability of hypothesis H being true. This is known as the prior probability.

P (E) is the probability of the evidence.

P (E|H) is the probability of the evidence given that the hypothesis is true.

P (H|E) is the probability of the hypothesis given that the evidence is there.

We classify unknown samples using naive Bayes classifier using a set workflow that is defined below:

1. We calculate a prior probability for given class labels.

2. Then we calculate the conditional probability with each feature for each class.

3. Then we multiple same class conditional probability 

4. Then multiply prior probability by the probability obtained in step 3

5. Then find the class with maximum probability and assign it to the given value.

Yes. Probability and statistics are two main pillars of theoretical computer science and any data science professional must be thorough on both to develop any sort of model for any data problem.

NB has the Following Pros:

1. It is very simple, easy, and fast to do binary and even multi-class prediction.

2. NB's performance increases manifold when the feature data set is independent.

3. Naive Bayes classifier performs better compared to other models like logistic regression for a balance d data set and NB also needs less training data for optimal performance.

4. For categorical data i.e. {'male', 'female', 'transgender'}, it performs better as compared to numerical data among the entire set of supervised learning-based classifiers. 

NB has the following cons:

1. NB fails to predict correctly if a categorical variable has a category (in the test data set), which was not observed in the training data set. In this case, the model will assign a 0 (zero) probability and will be unable to make a prediction. This is often known as “Zero Frequency”. To solve this, we can use the smoothing technique. One of the simplest smoothing techniques is called Laplace estimation.

2. NB is not a good probability estimator.

3. The major limitation of Naive Bayes is the assumption of independent predictors. In real life, it is almost impossible that we get a set of predictors which are completely independent.

The following are some of the major applications of the Naive Bayes algorithm.

1. Real-time Prediction

2. Multi-class Prediction

3. Text classification/ Spam Filtering/ Sentiment Analysis

4. Review and Recommendation System

5. Content moderation system

6. Market research

Yes, learning the Naïve Bayes classifier is good for your career on account of many factors. First, since it is based on the principle of probability, you will learn probability theory. Secondly, you will also learn the basics of supervised learning. Thirdly, you will learn generic principles of classification and especially text classification while learning Naïve Bayes. All these things are needed in various data science projects.

The easiest way to implement a Naive Bayes classifier is to use Python's sci-kit-learn library. This library has three types of Naive Bayes models that are based on the type of input data.

1. Gaussian: This is used for classification when the data is numerical and follows a normal distribution.

2. Multinomial: This is used for classification among more than two classes. 

3. Bernoulli: This is used for binary classification.

In general, there is no single "best" option for every classification problem.  But Naive Bayes is highly accurate when applied to big data. 

1. Naive Bayes is a linear classifier that works very fast when applied to big data.  Other supervised learning classifiers, such as K-NN, are usually slower for large amounts of data. This is because all of them had to perform large amounts of calculations at each new step in the process. If speed is important, Naive Bayes scores over all other options.

2. Naive Bayes offers you two hyperparameters to achieve smoothening i.e., alpha and beta. A hyperparameter is a prior parameter that is tuned on the training set to optimize it. Other methods have more hyperparameters that need to be tuned to reach optimality.

3. Naive Bayes has no effect due to the curse of dimensionality and large feature sets. Other methods such as K-NN have problems with both.

4. For tasks like robotics and computer vision, Bayes outperforms decision trees.