1. Understanding the Grid Setup

The problem describes a $6 \times 6$ square grid, which is composed of $6 \times 6 = 36$ small squares. The points of intersection, labeled $A_1, A_2, \ldots, A_{49}$, are the vertices of these small squares. A grid of $n \times n$ squares has $(n+1) \times (n+1)$ intersection points. For a $6 \times 6$ grid, this means we have $(6+1) \times (6+1) = 7 \times 7 = 49$ points. These points are arranged in a $7 \times 7$ lattice.

2. Calculating the Total Number of Outcomes

We are choosing two distinct points randomly from the 49 available points. The total number of ways to select two distinct points is the number of combinations of 49 items taken 2 at a time, denoted as $\binom{49}{2}$.

$\text{Total outcomes} = \binom{49}{2} = \frac{49!}{2!(49-2)!} = \frac{49 \times 48}{2} = 49 \times 24 = 1176$

3. Counting the Favorable Outcomes

A favorable outcome is a pair of points that are "friends". Two points are friends if they are adjacent along a row or a column. We can count the total number of such pairs by counting the horizontal and vertical adjacent pairs separately.

• Horizontal Pairs: The grid has 7 rows. In each row, there are 7 points. The number of adjacent pairs in a single row is $7 - 1 = 6$. For example, in the first row, the pairs are $(A_{1,1}, A_{1,2}), (A_{1,2}, A_{1,3}), \ldots, (A_{1,6}, A_{1,7})$. Since there are 7 rows, the total number of horizontal friendly pairs is: $N_{\text{horizontal}} = 7 \text{ rows} \times 6 \text{ pairs/row} = 42$

• Vertical Pairs: Similarly, the grid has 7 columns. In each column, there are 7 points. The number of adjacent pairs in a single column is $7 - 1 = 6$. Since there are 7 columns, the total number of vertical friendly pairs is: $N_{\text{vertical}} = 7 \text{ columns} \times 6 \text{ pairs/column} = 42$

The total number of favorable outcomes is the sum of the number of horizontal and vertical pairs, as these two sets of pairs are disjoint.

$\text{Favorable outcomes} = N_{\text{horizontal}} + N_{\text{vertical}} = 42 + 42 = 84$

4. Calculating the Probability, $p$

The probability $p$ is the ratio of the number of favorable outcomes to the total number of outcomes.

$p = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{84}{1176}$

To simplify the fraction:

$p = \frac{84}{49 \times 24} = \frac{12 \times 7}{7 \times 7 \times 24} = \frac{12}{7 \times 24} = \frac{1}{7 \times 2} = \frac{1}{14}$

5. Calculating the Final Value

The question asks for the value of $7p$.

$7p = 7 \times p = 7 \times \frac{1}{14} = \frac{7}{14} = \frac{1}{2} = 0.5$

Thus, the value of $7p$ is 0.5.