Probability of Two Chosen Grid Points Being Friends

To find the value of $7p$, we first calculate the total number of ways to choose two distinct points from the grid and the number of favorable pairs of points that are "friends".

1. Total Number of Points: A $6 \times 6$ square grid consists of $7$ horizontal lines and $7$ vertical lines. Thus, the total number of grid intersection points is: $N = 7 \times 7 = 49$

2. Total Sample Space: The total number of ways to choose any $2$ distinct points randomly out of these $49$ points is given by: $\text{Total ways} = \binom{49}{2} = \frac{49 \times 48}{2} = 49 \times 24 = 1176$

3. Number of Favorable Pairs ("Friends"): Two points are friends if they are adjacent along a row or a column (i.e., they form an edge in the grid graph).

   • Horizontal adjacent pairs: Each of the $7$ rows contains $7$ points, which gives $7 - 1 = 6$ adjacent pairs per row. $\text{Total horizontal pairs} = 7 \times 6 = 42$
   • Vertical adjacent pairs: Each of the $7$ columns contains $7$ points, which gives $7 - 1 = 6$ adjacent pairs per column. $\text{Total vertical pairs} = 7 \times 6 = 42$

   Therefore, the total number of favorable pairs is: $\text{Favorable pairs} = 42 + 42 = 84$

4. Calculating the Probability $p$: The probability $p$ that two randomly chosen distinct points are friends is: $p = \frac{\text{Favorable pairs}}{\text{Total ways}} = \frac{84}{1176} = \frac{84}{49 \times 24}$

   Simplifying the fraction: $p = \frac{12}{7 \times 24} = \frac{1}{7 \times 2} = \frac{1}{14}$

5. Value of $7p$: $7p = 7 \times \frac{1}{14} = \frac{1}{2} = 0.5$

Final Answer: The value of $7p$ is $0.5$.