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Probability on a Grid

  1. Nov 29, 2007 #1
    Consider a 3-by-3 square grid. Suppose you pick 5 of the squares at random. What is the probability that at least 1 line of 3 squares is formed? (3 diagonal squares is NOT a line).

    I do know that there are 6 combinations of lines that are possible. After that, I'm not sure how to follow along with this.
  2. jcsd
  3. Nov 29, 2007 #2


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    number of possible lines/number of possible ways you can select 3 out of 9
  4. Nov 29, 2007 #3
    I considered that, but you are not choosing 3 squares at a time... but instead, you are choosing a group of 5 squares. And from these 5 squares, what is the probability that there is a "line of 3 squares".
  5. Nov 29, 2007 #4
    The fact that [tex]_9C_5=126[/tex] shows that there are a total of 126 possible ways a group of five squares can be selected at random. Now how many of these 126 arrangements of squares contain at least one "line of 3 squares"? Do I have to draw all the possibilities (and consider some of the possibilities can be rotated or reflected to get another possibility)?

    So far I have found 49 of them (a total of 10 diagram arrangements which can occur multiple times with rotations or reflections). Should I continue this way, or is there a simpler way?
  6. Nov 29, 2007 #5


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    If the line goes up a side, top or bottom it's probably always a rotation of a form where it goes up the right

    note you can only have one line. So you put the line up the right, you have 6 spaces for 2 squares, multiply by 4 for rotation. Then you have to consider if the line goes up the middle
  7. Nov 29, 2007 #6
    Thank you very much for the insight Office_Shredder. Now I can make only the possible arrangements, and the remaining arrangements of the 126 will be the ones that don't create a "line of 3 squares".
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