Explanation:
To find the probability that the crosses form a horizontal, vertical, or diagonal line, we need to determine the total number of possible outcomes and the number of favorable outcomes.
In this case, we have a 3x3 grid, and we need to place three crosses randomly.
Total number of possible outcomes:
To calculate the total number of possible outcomes, we can consider placing the crosses one by one in the empty squares. For the first cross, there are 9 available squares. For the second cross, there are 8 remaining squares, and for the third cross, there are 7 remaining squares. So the total number of possible outcomes is 9 * 8 * 7 = 504.
Number of favorable outcomes:
To form a horizontal, vertical, or diagonal line, the crosses need to be placed in specific positions. There are 8 possible lines: 3 horizontal lines, 3 vertical lines, and 2 diagonal lines.
For horizontal lines, there are three options: top row, middle row, or bottom row.
For vertical lines, there are three options: left column, middle column, or right column.
For diagonal lines, there are two options: from top-left to bottom-right or from top-right to bottom-left.
In each case, there is only one way to arrange the crosses to form the desired line.
So the number of favorable outcomes is 3 (horizontal lines) + 3 (vertical lines) + 2 (diagonal lines) = 8.
Now we can calculate the probability:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 8 / 504
Probability = 1 / 63
Therefore, the probability that the crosses form a horizontal, vertical, or diagonal line is 1/63.