Question:
A rational number is selected at random from the distinct rational numbers of the form \( \frac{p}{q} \) formed with \( p \) and \( q \) belonging to the set \( \{1,2,3,4,5,6\} \). The probability that the rational number selected is a proper fraction is:
A rational number is selected at random from the distinct rational numbers of the form \( \frac{p}{q} \) formed with \( p \) and \( q \) belonging to the set \( \{1,2,3,4,5,6\} \). The probability that the rational number selected is a proper fraction is:
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To determine proper fractions in probability problems, count all valid numerators \( p<q \) and divide by total fraction possibilities.
Updated On: Jun 5, 2025
- \( \frac{1}{2} \)
- \( \frac{5}{6} \)
- \( \frac{11}{23} \)
- \( \frac{13}{35} \)
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The Correct Option is B
Solution and Explanation
Total distinct rational numbers:
\[
\text{Total selections} = 6 \times 6 = 36
\]
A proper fraction satisfies \( p<q \). The number of valid cases:
\[
\sum_{q=2}^{6} (q - 1) = 5 + 4 + 3 + 2 + 1 = 15
\]
Thus, probability:
\[
P(\text{proper fraction}) = \frac{30}{36} = \frac{5}{6}
\]
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