Mathematics | Conditional Probability

Conditional probability P(A | B) indicates the probability of event ‘A’ happening given that event B happened.

    P(A|B) = \frac{P(A \cap B)}{P(B)}

We can easily understand the above formula using the below diagram. Since B has already happened, the sample space reduces to B. So the probability of A happening becomes P(A \cap B) divided by P(B)

conditional_probab



Example:
In a batch, there are 80% C programmers, and 40% are Java and C programmers. What is the probability that a C programmer is also Java programmer?

Let A --> Event that a student is Java programmer
    B --> Event that a student is C programmer
    P(A|B) = P(A ∩ B) / P(B)
           = (0.4) / (0.8)
           = 0.5
So there are 50% chances that student that knows C also 
knows Java 



Product Rule:
Derived from above definition of conditional probability by multiplying both sides with P(B)




   P(A ∩ B) = P(B) * P(A|B) 


Understanding Conditional probability through tree:
Computation for Conditional Probability can be done using tree, This method is very handy as well as fast when for many problems.

Example: In a certain library, twenty percent of the fiction books are worn and need replacement. Ten percent of the non-fiction books are worn and need replacement. Forty percent of the library’s books are fiction and sixty percent are non-fiction. What is the probability that a book chosen at random are worn? Draw a tree diagram representing the data.

Solution: Let F represents fiction books and N represents non-fiction books. Let W represents worn books and G represents non-worn books.

P(worn)= P(N)*P(W | N) + P(F)*P(W | F)
              
           = 0.6*0.1  +  0.4* 0.2      
           = 0.14

Exercise:
1) What is the value of P(A|A)?

2) Let P(E) denote the probability of the event E. Given P(A) = 1, P(B) = 1/2, the values of P(A | B) and P(B | A) respectively are (GATE CS 2003)
(A) 1/4, 1/2
(B) 1/2, 1/14
(C) 1/2, 1
(D) 1, 1/2
See this for solution.



References:
http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-19-conditional-probability/

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