Probability, Theorems on Multiplication Law of Probability and Conditional Probability, Introduction to Bayes'Theorem, Theorem of Total Probability

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Theorems on Multiplication Law of Probability and Conditional Probability

Theorem 1:

For two events and ,

And

Where represents the conditional probability of occurrence of , when the event has already occurred and is the conditional probability of happening of , given that has already happened.

Proof:

Let denote the total number of equally likely cases, denote the cases favourable to the event , denote the cases favourable to and denote the cases favourable to both and .

For the conditional event , the favourable outcomes must be one of the sample points of , i.e., for the event , the sample space is and out of the sample points, pertain to the occurrence of the event , Hence,

Rewriting (1), we get

Similarly, we can prove

Theorem 2:

Two events and of the sample space are independent, if and only if

Proof:

If and are independent events,

Then

We know that

Hence, if and are independent events, then the probability of ‘ and ’ is equal to the product of the probability of and probability of .

Conversely, if , then

gives

That is, and are independent events.

Introduction to Bayes’ Theorem

  • In conditional probability we have learnt to find probability of an event with the condition that some other event has already occurred. Consider an experiment of selecting one coin out of three coins: If I with and , II with and and III with , (a normal coin).

  • After randomly selecting one of the coins, it is tossed. We can find the probability of selecting one coin and can also find the probability of any outcome i.e. head or tail; given the coin selected. But can we find the probability that coin selected is coin I, II or III when it is known that the head occurred as outcome? For this we have to find the probability of an event which occurred prior to the given event. Such probability can be obtained by using Bayes’ theorem, named after famous mathematician, Johan Bayes Let us first learn some basic definition before taking up Baye’s theorem

Mutually exclusive and exhaustive events.

For a sample space, the set of events is said to mutually exclusive and exhaustive if

(i) i.e. none of two events can occur together.

(ii) , all outcomes of S have been taken up in the events

(iii) for all

Theorem of Total Probability

Let are mutually exclusive and exhaustive events for a sample space with . Let be any event associated with , then

Proof:

The events and are shown in the Venn-diagram

Given and

We can write

Since all, are mutually exclusive, so will also be mutually exclusive

By using the multiplication rule of probability,

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