What is the difference between conditional probability and probability?

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In short, “Each conditional probability is a probability and the converse is also true.”

The 1st part of the above statement is obvious but how is the converse true? To understand things better, let us first revisit our definition of probability of an event A. Let S be the set of all possible outcomes of a process and A (favourable event) be the set of all points of our interest from S. Then  P(A) is defined as

where n(A) is the number of points in A and n(S) is the number of points in S. (For more details on Probability visit my earlier post….).

Example 1: If I ask, what is the probability of getting a number which is a multiple of 3, when you throw a dice. You immediately answer it as 1/3. Because the favourable event A = {3,6}, sample space S1 = {1,2,3,4,5,6}, n(A)=2 and n(S) = 6. 

But if I ask you the same question, what is the probability of picking a number which is a multiple of 3? You immediately ask me “From where should I pick a number?”, i.e., indirectly you are asking me, “Tell me the sample space”.

Example 2: If I say that the sample space is the number set S2 = {1,2,3,4}. Then you will answer P(A) = 1/4.  

Example 3: If I say that the sample space is the number set  S3 = {1,2,3,4,5}. Then you will answer P(A) = 1/5. 

What have you noticed from the above three examples? Each time I asked you the same question, but you gave me different answers. Why?

Because each time I am changing the sample space and your answer is also changing accordingly. That is the point I want to make here. When you are talking about probability P(A), you actually mean probability of an event A, given the sample space is S. In mathematical language, we write it P(A|S), which is nothing but the conditional probability of A when you observe S as your sample space. Thus, 

in example 1: P(A) = P(A|S1) = 1/3, 

in example 2: P(A) = P(A|S2) = 1/4,

In example 3: P(A) = P(A|S3) = 1/5. 

Now you understand that the probability of an event A is nothing but a conditional probability of A given the sample space S.  In practice, we use the notation P(A) instead of P(A|S), when S is the whole sample space or universal sample space. And we use the notation P(A|B) when B is a subset of S.

The question arises, how to calculate P(A|B) when B is a subset of S?

Okay, the answer is very easy. Consider B as your new sample space. Find n’(A), the number of points from B that are satisfying the statement of interest A and then

where n(B) is the number of points in B. Note that when B = S, P(A|B) becomes P(A).

Also, notice that n’(A), number of points in A when we consider B as our sample space, is same as n(AB), number of points in AB when we consider S as our sample space. 

Thus we can also express (2) as

Example 4: Consider S is the set of all single-digit non zero numbers. Then S = {1,2,3,4,5,6,7,8,9}. Let A is the event that contains numbers which are a multiple of three and B is the event that contains numbers which are less than 6, i.e., A = {3,6,9} and B =.{1,2,3,4,5}. Note that AB = {3}. Then P(A) = 1/3,  P(B) = 5/9 and P(AB)= 1/9. We now calculate P(A|B) using both the formulae (2) and (3). We see that B contains only 5 elements of which only one element is satisfying  A. Using (2), we get 

If we use the formula (3), we get 

Thus from this article, we understand that each probability is some kind of conditional probability. We also learn when to use the term conditional probability and when to use the term probability. Finally, we see that we can calculate conditional probability in two different ways and arrive at the same value.

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