The squares of a 3×3 board are randomly coloured black or white. Let W be the number of white squares and B the number of black squares. The expected value of |B−W| can be expressed as ab where a and b are coprime positive integers. What is the value of a+b?

To find the expected value of |B-W|, we need to consider all possible arrangements of coloring the squares of the 3x3 board and calculate the absolute difference between the number of black squares (B) and the number of white squares (W) for each arrangement.

There are a total of 2^9 = 512 possible arrangements of coloring the squares because each square can be either black or white, giving 2 options, and there are 9 squares in total.

To calculate the expected value, we need to find the sum of all absolute differences and divide it by the total number of arrangements.

Let's consider each possible number of black squares separately:

1. If there are exactly 0 black squares, then there are 9 white squares. The absolute difference is |0 - 9| = 9. There is only one arrangement with all white squares in this case.

2. If there is exactly 1 black square, then there are 8 white squares. The absolute difference is |1 - 8| = 7. There are 9 possible arrangements with 1 black square.

3. If there are exactly 2 black squares, then there are 7 white squares. The absolute difference is |2 - 7| = 5. There are (9 choose 2) = 36 possible arrangements with 2 black squares.

4. If there are exactly 3 black squares, then there are 6 white squares. The absolute difference is |3 - 6| = 3. There are (9 choose 3) = 84 possible arrangements with 3 black squares.

5. If there are exactly 4 black squares, then there are 5 white squares. The absolute difference is |4 - 5| = 1. There are (9 choose 4) = 126 possible arrangements with 4 black squares.

6. If there are exactly 5 black squares, then there are 4 white squares. The absolute difference is |5 - 4| = 1. There are (9 choose 5) = 126 possible arrangements with 5 black squares.

7. If there are exactly 6 black squares, then there are 3 white squares. The absolute difference is |6 - 3| = 3. There are (9 choose 6) = 84 possible arrangements with 6 black squares.

8. If there are exactly 7 black squares, then there are 2 white squares. The absolute difference is |7 - 2| = 5. There are (9 choose 7) = 36 possible arrangements with 7 black squares.

9. If there are exactly 8 black squares, then there is 1 white square. The absolute difference is |8 - 1| = 7. There are 9 possible arrangements with 8 black squares.

10. If there are exactly 9 black squares, then there are 0 white squares. The absolute difference is |9 - 0| = 9. There is only one arrangement with all black squares in this case.

Now, we can calculate the expected value:

Expected value = (9/512)*9 + (9/512)*7 + (36/512)*5 + (84/512)*3 + (126/512)*1 + (126/512)*1 + (84/512)*3 + (36/512)*5 + (9/512)*7 + (1/512)*9
= 81/512 + 63/512 + 180/512 + 252/512 + 126/512 + 126/512 + 252/512 + 180/512 + 63/512 + 9/512
= 1221/512

Therefore, the expected value of |B-W| is 1221/512.

The value of a+b = 1221+512 = 1733.

To find the expected value of |B−W|, we need to consider all possible colorings of the squares and determine the number of occurrences for each value of |B−W|.

We can start by listing all the possible colorings of the 3×3 board:

Black: 9, White: 0
Black: 8, White: 1
Black: 7, White: 2
Black: 6, White: 3
Black: 5, White: 4
Black: 4, White: 5
Black: 3, White: 6
Black: 2, White: 7
Black: 1, White: 8
Black: 0, White: 9

We will calculate the probability of each coloring occurring first:

For Black: 9, White: 0: The probability is (1/2)^9 = 1/512
For Black: 8, White: 1: The probability is (9/512)
For Black: 7, White: 2: The probability is (36/512)
For Black: 6, White: 3: The probability is (84/512)
For Black: 5, White: 4: The probability is (126/512)
For Black: 4, White: 5: The probability is (126/512)
For Black: 3, White: 6: The probability is (84/512)
For Black: 2, White: 7: The probability is (36/512)
For Black: 1, White: 8: The probability is (9/512)
For Black: 0, White: 9: The probability is (1/512)

Next, we will calculate the value of |B−W| for each coloring:

For Black: 9, White: 0: |B−W| = 9
For Black: 8, White: 1: |B−W| = 7
For Black: 7, White: 2: |B−W| = 5
For Black: 6, White: 3: |B−W| = 3
For Black: 5, White: 4: |B−W| = 1
For Black: 4, White: 5: |B−W| = 1
For Black: 3, White: 6: |B−W| = 3
For Black: 2, White: 7: |B−W| = 5
For Black: 1, White: 8: |B−W| = 7
For Black: 0, White: 9: |B−W| = 9

Now, we can calculate the expected value:

Expected value = (9 * (1/512) + 7 * (9/512) + 5 * (36/512) + 3 * (84/512) + 1 * (126/512) + 1 * (126/512) + 3 * (84/512) + 5 * (36/512) + 7 * (9/512) + 9 * (1/512))

Simplifying this expression, we get:

Expected value = (1/512) * (9 + 7 * 9 + 5 * 36 + 3 * 84 + 1 * 126 + 1 * 126 + 3 * 84 + 5 * 36 + 7 * 9 + 9)

Expected value = 90/512

Therefore, a = 90 and b = 512.

The final answer is a + b = 90 + 512 = 602.