Consider an experiment of tossing two fair dice, one black and one red. What is the probability that the number on the black die divides the number on red die ?
\(\frac{14}{36}\)
The sample space has \(6 \times 6 = 36\) equally likely ordered pairs (black, red); count the pairs where the black value divides the red value.
Black = 1 divides all 6 values; black = 2 divides 2, 4, 6 (3 cases); black = 3 divides 3 and 6 (2 cases).
Black = 4, 5 and 6 each divide only themselves, giving 1 case each.
Favourable \(= 6 + 3 + 2 + 1 + 1 + 1 = 14\), so probability \(= \frac{14}{36}\)
Check: \(\sum_{b=1}^{6} \lfloor 6/b \rfloor = 6+3+2+1+1+1 = 14\).
Hence, \(\frac{14}{36}\).
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