Probability-7 (IIT-2003)

For a student to qualify a competitive exam, he must pass at least two out of the three exams. The probability that he will pass the first exam is p. If he fails in one of the exams, then the probability of his passing in the next exam is p/2, otherwise it remains the same. Find the probability that the student will qualify the competitive exam.  

Solution follows here:

Solution:

For qualifying in the competitive exam, he has to pass either all the three exams or any two. The following table summarizes all the possibilities:

The conditions to be taken care here:

1.      “If he fails in one of the exams, then the probability of his passing in the next exam is p/2, otherwise it remains the same”.

2.      If probability of passing in an exam is ‘p’, then probability of failing is ‘1-p’.

3.      Passing/Failing in any exam is independent of the result of other two exams

=> P(E) = P(E₁)* P(E₂)* P(E₃)

Exam-I	Exam-II	Exam-III	Probability =P(E1)* P(E2)* P(E3)
p	p	p	p*p*p = p3
p	p	(1-p)	p*p*(1-p) = p2-p3
p	(1-p)	p/2	p*(1-p)*(p/2) = (p2/2)-(p3/2)
(1-p)	p/2	p	(1-p)*(p/2)*p = (p2/2)-(p3/2)

(Note that, in the above table, red colour indicates failure and green indicates passing in the respective exam)

Net Probability = p³+p²-p³+(p²/2)-(p³/2)+ (p²/2)-(p³/2) = 2p²-p³