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:
Solution follows here:
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 ‘1p’.
3.
Passing/Failing
in any exam is independent of the result of other two exams
=> P(E) = P(E_{1})* P(E_{2})* P(E_{3})
ExamI

ExamII

ExamIII

Probability
=P(E_{1})* P(E_{2})* P(E_{3})

p

p

p

p*p*p = p^{3}

p

p

(1p)

p*p*(1p)
= p^{2}p^{3}

p

(1p)

p/2

p*(1p)*(p/2)
= (p^{2}/2)(p^{3}/2)

(1p)

p/2

p

(1p)*(p/2)*p
= (p^{2}/2)(p^{3}/2)

(Note
that, in the above table, red colour indicates failure and green indicates passing
in the respective exam)
Net
Probability = p^{3}+p^{2}p^{3}+(p^{2}/2)(p^{3}/2)+
(p^{2}/2)(p^{3}/2) = 2p^{2}p^{3}
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