Key Concepts and Formulas

• Classical Definition of Probability: The probability of an event $E$ is the ratio of the number of favorable outcomes to the total number of possible outcomes, assuming all outcomes are equally likely. $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
• Complementary Events: If $E$ is an event, then its complement, denoted as $E'$, is the event that $E$ does not occur. The sum of the probabilities of an event and its complement is always 1. This relationship is often used to simplify calculations, especially when counting the favorable outcomes for $E'$ is easier than for $E$. $P(E) = 1 - P(E')$
• Combinations: When selecting $r$ items from a set of $n$ distinct items, where the order of selection does not matter, the number of ways to do this is given by the combination formula: ${}^{n}C_r = \frac{n!}{r!(n-r)!}$

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Step-by-Step Solution

Step 1: Calculate the Total Number of Possible Ways to Choose Two Squares

First, we need to determine the total number of distinct pairs of squares that can be chosen from the 16 squares on the board.

• Reasoning: We are selecting 2 squares out of 16. The order in which we pick the squares does not change the pair itself (e.g., choosing square A then square B is the same pair as choosing square B then square A). Therefore, this is a combination problem.
• Calculation: Using the combination formula with $n=16$ (total squares) and $r=2$ (squares to choose): $\text{Total possible outcomes} = {}^{16}C_2$ ${}^{16}C_2 = \frac{16!}{2!(16-2)!} = \frac{16!}{2!14!} = \frac{16 \times 15}{2 \times 1}$ ${}^{16}C_2 = 8 \times 15 = 120$ Thus, there are 120 unique ways to choose two squares from the $4 \times 4$ board. This will be the denominator for our probability calculation.

Step 2: Identify and Count Pairs of Squares with a Common Side (Complementary Event)

The problem asks for the probability that two chosen squares have no side in common. It is often simpler to count the number of outcomes for the complementary event: that the two chosen squares do have a common side (i.e., they are adjacent). Let's call this event $E'$.

• Reasoning: Two squares have a common side if they are immediately next to each other, either horizontally or vertically. We need to systematically count all such pairs on a $4 \times 4$ grid to ensure accuracy.
• Calculation:
  1. Horizontal Adjacent Pairs: Consider each row. In a row of 4 squares, there are $4-1=3$ pairs of adjacent squares. Since there are 4 such rows on the board: $\text{Number of horizontal adjacent pairs} = 4 \text{ rows} \times 3 \text{ pairs/row} = 12$
  2. Vertical Adjacent Pairs: Similarly, consider each column. In a column of 4 squares, there are $4-1=3$ pairs of adjacent squares. Since there are 4 such columns on the board: $\text{Number of vertical adjacent pairs} = 4 \text{ columns} \times 3 \text{ pairs/column} = 12$ The total number of pairs of squares that have a common side is the sum of horizontal and vertical adjacent pairs. These two types of pairs are mutually exclusive, so we simply add them. $\text{Number of outcomes for } E' = 12 + 12 = 24$

Step 3: Calculate the Probability of the Complementary Event ($P(E')$)

Now we can calculate the probability that the two randomly chosen squares do share a common side.

• Reasoning: This is the probability of our complementary event $E'$. We use the classical definition of probability with the number of favorable outcomes for $E'$ (calculated in Step 2) and the total possible outcomes (calculated in Step 1).
• Calculation: $P(E') = P(\text{common side}) = \frac{\text{Number of pairs with a common side}}{\text{Total number of possible pairs}} = \frac{24}{120}$ Simplifying the fraction: $P(\text{common side}) = \frac{24}{120} = \frac{1}{5}$

Step 4: Calculate the Desired Probability using Complementary Probability ($P(E)$)

Finally, we use the principle of complementary probability to find the desired probability: that the two chosen squares have no common side.

• Reasoning: The event "two chosen squares have no common side" ($E$) is the complement of the event "two chosen squares have a common side" ($E'$). Therefore, we subtract the probability of the complementary event from 1.
• Calculation: $P(E) = P(\text{no common side}) = 1 - P(\text{common side})$ $P(\text{no common side}) = 1 - \frac{1}{5}$ $P(\text{no common side}) = \frac{5}{5} - \frac{1}{5} = \frac{4}{5}$

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Common Mistakes & Tips

• Leverage Complementary Probability: Always consider if calculating the probability of the complementary event is simpler. This is particularly true for problems involving "no common," "at least one," or "not together" conditions, as it often reduces the complexity of direct counting.
• Systematic Counting for Grid Problems: When counting specific configurations on a grid (like adjacent squares), adopt a systematic approach to avoid miscounting. For an $m \times n$ grid:
  • Number of horizontal adjacent pairs = $m \times (n-1)$
  • Number of vertical adjacent pairs = $(m-1) \times n$ For our $4 \times 4$ grid, this yields $4 \times (4-1) = 12$ horizontal pairs and $(4-1) \times 4 = 12$ vertical pairs, totaling 24.
• Combinations vs. Permutations: Clearly distinguish between combinations (${}^n C_r$) and permutations (${}^n P_r$). Use combinations when the order of selection does not matter (as in choosing a pair of squares), and permutations when order is significant (e.g., arranging items in a specific sequence).

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Summary

To determine the probability that two randomly chosen squares from a $4 \times 4$ board have no side in common, we utilized the strategy of complementary probability. First, we calculated the total number of ways to choose any two squares from the 16 available squares using combinations, finding ${}^{16}C_2 = 120$. Next, we identified the complementary event: choosing two squares that do have a common side. By systematically counting horizontal and vertical adjacent pairs, we determined there were $12 + 12 = 24$ such pairs. This allowed us to calculate the probability of the complementary event as $P(\text{common side}) = \frac{24}{120} = \frac{1}{5}$. Finally, subtracting this from 1 gave us the desired probability: $P(\text{no common side}) = 1 - \frac{1}{5} = \frac{4}{5}$.

The final answer is $\boxed{\frac{4}{5}}$, which corresponds to option (A).