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When we should use Total probability and Baye's Theorem?

Explain with examples.

Total probability is meant for finding the probability of happening of an event due to all the possible causes of that event. And bayes theorem is used in the case when the event has occurred and you need to find the cause due to which this event has occurred from all the possible causes.

Now, let's take an example, suppose we have an event z, and we have other two events A and B that acts as the cause of event z.

So, the total probability says,   P(z)= P(z|A)*P(A) +P(z|B)*P(B)

And now first understand the term P(z|A)*P(A). It means that , first the event A has occurred and now, after happening of event A only, the probability of occurring of event z is. Conditional probability shrinks the sample space from universal space to the event A only for event z.

So, that means the verbal statement for total probability equation is: " The probability of occurring of event z , if event A occurs + the probability of occurring of event z if event B occurs." . This statement gives the total probability of occurring of event z due to these two causes event A and event B.

Similarly, Bayes theorem handles the condition, when it is given that event z has occurred, and now we need to find the probability that due to which causing event from A and B, event z has occurred
Think of it this way.

Total probabilty, TOTAL implies that including all possible ways for a output to come. if X and Y are two possible ways,we sum them and write X+Y as the total probability. For calculating X and Y we might use " conditional probability in case of dependant events ".

Now Bayes theorem is actually interesting, it asks us this " what is the chance that this output might come from a source A?".

calculate probability for happening of certain event  considering all possible events-->  use TOTAL probability

Given the probabilty, what is the chance that this came from a certain Source?--> BAYES THEOREM

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