Probability
Probability
Allen Star Batch
Grade 12
Question:
Two friends decide to meet at a spot between 2 p.m. and 3 p.m. whosoever arrives first agrees to wait for 15 minutes for the other. The probability that they met is $\frac{p}{q}$ (where $H.C.F(p, q) = 1$) then $q - p = \ldots\ldots\ldots\ldots$
Step-by-Step Solution
Key Concept: Geometric probability involves finding the ratio of favorable area to total area in a coordinate plane representation.
Model arrival times of $A$ and $B$ along the $x$ and $y$ axes respectively (in minutes, scale 0-60). They meet if $|x - y| \leq 15$. The sample space is a $60 \times 60$ square with area $3600$. The favorable region where $|x-y| \leq 15$ is the band between lines $y = x - 15$ and $y = x + 15$. The unfavorable area consists of two corner triangles with total area $45^2 = 2025$. Probability = $\frac{60^2 - 45^2}{60^2} = \frac{3600 - 2025}{3600} = \frac{7}{16}$.
Correct Answer: 9