If the squares of a `8 xx 8` chess board are painted either red and black at random .The probability that not all squares is any alternating in colour is

Options

A. `1//2^(64)`
B. 1-`1//2^(63)`
C. `1//2`
D. none of these

Explanation

To solve the problem of finding the probability that not all squares of an 8x8 chessboard are alternating in color when painted randomly red or black, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Chessboard Configuration**:
- An 8x8 chessboard consists of 64 squares.
- Each square can be painted either red or black.

2. **Total Possible Colorings**:
- Since each of the 64 squares can be painted in 2 different colors (red or black), the total number of ways to paint the chessboard is:
\[ \text{Total Outcomes} = 2^{64} \]

3. **Favorable Outcomes for Alternating Colors**:
- An alternating color pattern means that no two adjacent squares can be of the same color. There are two possible alternating patterns:
- Pattern 1: Starting with Red (R):
- R B R B R B R B
- B R B R B R B R
- Pattern 2: Starting with Black (B):
- B R B R B R B R
- R B R B R B R B
- Thus, there are 2 favorable outcomes for the chessboard to be painted in an alternating color pattern.

4. **Calculating the Probability of Alternating Colors**:
- The probability \( P \) that all squares are in an alternating color pattern is given by:
\[ P(\text{alternating}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{2}{2^{64}} = \frac{1}{2^{63}} \]

5. **Finding the Probability of Not All Squares Being Alternating**:
- The probability that not all squares are in an alternating color pattern is the complement of the probability that they are in an alternating pattern:
\[ P(\text{not alternating}) = 1 - P(\text{alternating}) = 1 - \frac{1}{2^{63}} \]

6. **Final Result**:
- Therefore, the probability that not all squares are in an alternating color is:
\[ P(\text{not alternating}) = 1 - \frac{1}{2^{63}} \]

### Summary of the Solution:
The final answer is:
\[ P(\text{not alternating}) = 1 - \frac{1}{2^{63}} \]