Probability of Forming a Line on a 3x3 Grid with Random Selections

In summary, the conversation discusses the probability of forming at least one line of 3 squares in a 3-by-3 square grid by randomly selecting 5 squares. There are 6 possible combinations of lines and 126 possible ways to select 5 squares. After considering rotations and reflections, it is determined that there are 49 arrangements with a line of 3 squares, leaving 77 arrangements without a line. The final probability is 49/126 or approximately 0.39.
  • #1
atqamar
55
0
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.
 
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  • #2
number of possible lines/number of possible ways you can select 3 out of 9
 
  • #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".
 
  • #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?
 
  • #5
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
 
  • #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".
 

1. What is the definition of probability on a grid?

Probability on a grid is a mathematical concept that refers to the likelihood or chance of an event occurring on a coordinate grid. It is a way of quantifying uncertainty and is often used to analyze and predict outcomes in various fields such as physics, economics, and engineering.

2. How is probability calculated on a grid?

The probability on a grid is calculated by dividing the number of successful outcomes by the total number of possible outcomes. For example, if there are 4 red squares and 6 blue squares on a grid, the probability of landing on a red square would be 4/10 or 40%.

3. What is the difference between discrete and continuous probability on a grid?

Discrete probability on a grid refers to situations where the possible outcomes are countable and distinct, such as rolling a dice or flipping a coin. Continuous probability on a grid, on the other hand, involves measuring outcomes on a continuum, such as the height of a person or the time it takes to complete a task.

4. How is probability used in real life applications on a grid?

Probability on a grid has a wide range of real-life applications, including predicting stock market trends, weather forecasting, and risk analysis in insurance. It is also used in games of chance, such as casino games and lottery draws.

5. What are some common misconceptions about probability on a grid?

One common misconception is that probability is solely based on chance, when in reality it also involves understanding and analyzing patterns and data. Another misconception is that past events can influence future outcomes, when in fact each event is independent and has its own probability of occurring.

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