Probability
Probability
Allen Star Batch
Grade 12
Question:
A student appears for test I, II and III. The student is successful if he passes either in test I, II or I, III. The probability of the student passing in test I, II and III are respectively $p$, $q$ and $1/2$. If the probability of the student to be successful is $1/2$ then :
$p = q = 1$
$p = q = \frac{1}{2}$
$p = 1, q = 0$
$p = 1, q = \frac{1}{2}$
Step-by-Step Solution
Key Concept: The student succeeds if passing in (Test I AND Test II) OR (Test I AND Test III). Using the inclusion-exclusion principle for P(A∪B) = P(A) + P(B) - P(A∩B) where A = I∩II and B = I∩III, combined with independence of test results, leads to the constraint equation that determines the unique solution.
Expanding $P(I \cup I \cup III) = P(I) + P(II) - P(I \cap II) - P(I \cap III) - P(II \cap III) + P(I \cap II \cap III)$ gives $pq + \frac{p}{2} - \frac{pq}{2} - \frac{p}{2}(q+1) + \frac{1}{2} = 1$, which simplifies to $p(q+1) = 1$.
Correct Answer: 3