Probability
PYP_JEE_ADV_2023_P2
Grade None
Question:
**PARAGRAPH \"II\"**\nConsider the $6 \times 6$ square in the figure. *[Image: A 6x6 square grid with 49 intersection points (dots) arranged uniformly]* Let $A_1, A_2, \dots, A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_i$ and $A_j$ are friends if they are adjacent along a row or along a column. Assume that each point $A_i$ has an equal chance of being chosen.\n\nTwo distinct points are chosen randomly out of the points $A_1, A_2, \dots, A_{49}$. Let $p$ be the probability that they are friends. Then the value of $7p$ is
Step-by-Step Solution
Key Concept: Counting edges in a grid graph and calculating probability by combining combinations.
**Step 1: Count the total number of pairs**
Two distinct points are chosen from the 49 points. The total number of possible pairs is $\binom{49}{2} = \frac{49 \times 48}{2} = 49 \times 24 = 1176$.
**Step 2: Count the number of friendly pairs**
Two points are friends if they form an edge of the grid graph. We need to count the total number of horizontal and vertical edges.\nThere are 7 rows of points, and each row has 6 horizontal segments connecting adjacent points. Total horizontal edges = $7 \times 6 = 42$.\nThere are 7 columns of points, and each column has 6 vertical segments. Total vertical edges = $7 \times 6 = 42$.\nTotal number of edges (friendly pairs) = $42 + 42 = 84$.
**Step 3: Calculate the probability**
The probability $p$ that the two chosen points are friends is the ratio of friendly pairs to total pairs:\n$p = \frac{84}{1176} = \frac{84}{49 \times 24} = \frac{7}{49 \times 2} = \frac{1}{14}$.
**Step 4: Calculate the final required value**
We need to find the value of $7p$.\n$7p = 7 \times \frac{1}{14} = \frac{1}{2} = 0.5$.
Correct Answer: 0.5