SATHEE: Probability 3 Question 36

Probability 3 Question 36

36. For a student to qualify, he must pass atleast two out of three exams. The probability that he will pass the 1st exam is $p$ . If he fails in one of the exams, then the probability of his passing in the next exam, is $\frac{p}{2}$ otherwise it remains the same. Find the probability that he will qualify.

(2003, 2M)

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Solution:

Let $E_{i}$ denotes the event that the students will pass the $i$ th exam, where $i = 1, 2, 3$

and $E$ denotes the student will qualify.

$\begin{aligned} ∴ P (E) = & [P (E_{1}) \times P (E_{2} / E_{1})] \\ + [P (E_{1}) \times P (E_{2}^{'} / E_{1}) \times P (E_{3} / E_{2}^{'})] \\ + [P (E_{1}^{'}) \times P (E_{2} / E^{'}_{1}) \times P (E_{3} / E_{2})] \\ = p^{2} + p (1 - p) \cdot \frac{p}{2} + (1 - p) \cdot \frac{p}{2} \cdot p \\ \Rightarrow & P (E) = \frac{2 p^{2} + p^{2} - p^{3} + p^{2} - p^{3}}{2} = 2 p^{2} - p^{3} \end{aligned}$

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Probability 3 Question 36

36. For a student to qualify, he must pass atleast two out of three exams. The probability that he will pass the 1st exam is p. If he fails in one of the exams, then the probability of his passing in the next exam, is p2 otherwise it remains the same. Find the probability that he will qualify.

Solution:

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