Probability Theory

Probability theory is a part of mathematics that analyses a random phenomenon. The outcome of any event cannot be exactly predicted before it actually occurs. But there are several possibilities of an outcome. A study of such ‘several possibilities’ is nothing but the probability theory. The word ’probability’ is synonymous to ‘possibility’ and hence the probability theory is a study of the various possibilities by which an even can occur.

Probability Theory Definition

Probability theory deals with uncertainty of any event or an random experiment.

Probability Theory and Examples

The simplest definition of a probability is the ratio of the number of possible outcomes to the total of all possible outcomes which is also called as sample space. The classic examples that are used to explain the concept of probability are, tossing a coin, rolling a dice, taking a card from the deck of playing cards. But the approach to the probability in each case depends on what exactly you are looking for. Tossing of a  single coin has only two possible outcomes, tossing ‘head’ or ‘tossing tail’, shortly describing as ‘H’ or ‘T’. The set of outcomes is [H, T]. Suppose two coins are tossed, then the set of outcomes changes to [HH, HT, TH, TT]. In other words, the sample space changes from 2 to 4. Similarly the sample space in a die is 6 and the favorable outcome is 1 when you want to experiment the probability of rolling on a particular number. But if you redefine your favorable outcome as ‘rolling on even number’ or ‘rolling on odd number’ the sample space is just 2.
Therefore, there can be different sample spaces for the same experiment and also the number of favorable outcomes can be defined in different ways.  A set of outcomes is defined as sample space and each element in the set of outcomes is called a sample point.   

The type of a probability is also dependent of the nature of events. An event is said to be simple if it has only one outcome.

For example, if you toss a coin the outcome can either be a ‘head’ or ‘tail’.

If an event can be a part of two outcomes, then it is called a compound outcome. For example, tossing at least one head in two tosses of a coin is a compound event. It is because it can happen with any of the possible outcomes HH, HT or TH.

If two events cannot happen at the same time, then the set of events is called mutually exclusive events. It is not possible to roll number 6 and a prime number by rolling a single die.
A set of events is called ‘exhaustive events’ if the probability of at least one of them is 1. As an example, suppose you roll a die for the following outcomes, A) Rolling a number greater than 3, B) Rolling a prime numbe,r C) Rolling an odd number. When the die is actually rolled, any of the outcomes is sure to happen.

Now let us discuss the most important and the revolutionary concept in probability theory.

In many cases, you are on the lookout for the probability of just ‘yes’ or ‘no’ or at least you converge the various possibilities into two.

For example, if your score is 5 out of 10 and if you are declared ‘pass’, then you look for the probability of ‘pass’ or ‘fail’, irrespective of various possibilities of actual scores. This type of probability is defined as ‘binomial probability’. Most of the probabilities in real life situations are binomial probabilities. Extensive trials are conducted on many important events and a formula has been derived known as ‘Bernoulli formula’ of binomial probability.
The study is further extended for different possible outcomes and the study is termed as ‘binomial distribution’.  It has been found that the probabilities worked out on the basis of binomial distribution are fairly close to probabilities in a normal distribution when the number of trials is large.The concept of binomial probability for exact number of successes is extended to realistic pattern of ‘almost’ and ‘at least’ the ‘number of successes’ which fits much better for practical situations.Let us take the example of tossing a coin 100 times. With the probability theory it is possible to evaluate the probability of tossing head (or tail) exactly, at most, at least, greater than or less than a particular number of times. Similarly, it is possible to find a probability like maximum how many persons can survive from a epidemic when a record on survival on such epidemics is known.       

Total Probability Theorem

This is very important theorem which helps in finding a complex probability when a multiple probabilities are known. This theorem is something like chain rule in differential calculus.
The theorem states that if A is an event and for any partition B1, B2, B3, ……Bn a for the same event, then,
P(A) = P(B1)*P(A/ B1) + P(B2)*P(A/ B2) + P(B3)*P(A/ B3) + ……….. + P(Bn)*P(A/ Bn)
Let us take an example in real life to illustrate this.

Suppose a television industry has factories in the states of California, Texas, Arizona and New Jersey manufacturing 200, 350, 150 and 300 units a day, respectively. The defective rates are 5% for California, 8% for Texas, 6% for Arizona and 10% for New Jersey. Suppose you a TV is purchased from a mall in New York, without knowing the place of manufacture, what is the probability that the TV will fail?
The total number of TVs manufactured in all the states = (200 + 350 + 150 + 300) = 1000 per day

The probability of the TV manufactured in California is, P(C) = (200)/1000) = 0.2   
The probability of the TV manufactured in Texas is, P(T) = (350)/1000) = 0.35
The probability of the TV manufactured in Arizona is, P(A) = (150)/1000) = 0.15
The probability of the TV manufactured in New Jersey is, P(N) = (300)/1000) = 0.3

The probability of a TV manufactured in California being defective is, P(D/C) = 5% = 0.05
The probability of a TV manufactured in Texas being defective is, P(D/T) = 8% = 0.08     
The probability of a TV manufactured in Arizona being defective is, P(D/C) = 6% = 0.06
The probability of a TV manufactured in New Jersey being defective is, P(D/N) = 10% = 0.10

Therefore, the total probability of a TV picked up at random being defective is,
P(D) =  P(C)*P(D/C) + P(T)*P(D/T) + P(A)*P(D/A) + P(N)*P(D/N)
        = 0.2*0.05 + 0.35*0.08 + 0.15*0.06 + 0.3*0.1 = 0.01 + 0.028 + 0.009 + 0.03 = 0.077 = 7.7%