Grid-based Probability

[Atticus837]

Suppose I have a 10x10 grid (or even nxn) and I fill in the squares one by one, in an order that is randomly determined. After how many squares is the probability of completing ANY row and/or column of the grid to be greater than 50%? In the 10x10 example, with 10 squares filled the probability is absurdly small that all 10 of them lie in the same row or column, but with 90 squares filled it's almost a certainty that there's a complete row or column somewhere on the grid. My intuition tells me the answer may be 50/100, but that seems like quite an unsatisfying answer. We're dealing with such large numbers of combinations here, but does anyone know how to approach a problem like this? Not for schoolwork or anything, just been bugging me.

 

[Steven G]

What have you tried? Where did you get stuck? Did you try this with smaller numbers? Maybe 2x2 and then 3x3 to see if you see a pattern. Please post back with your work. For the record you would have received help by now if you followed the forum guidelines which states in part that you need to show your work in order to get help.

 

[Atticus837]

My apologies. This isn't for school work or anything, I'm a math geek in his mid 20s. I suspect maybe this is the wrong site to be posting on.

But here goes:
I tried starting with smaller numbers, but it quickly becomes hard to calculate. I couldn't see a pattern.
I put together a spreadsheet, but here's where I'm having a problem.

2x2 is simple. When you fill 2 squares you have a 66.6...% probability of having a completed row or column. Any other value is trivial. (1 is 0%, 3 is 100%) 3x3 gets harder. For example, using simple combinations I worked out that with 3 squares filled the probability of a complete row or column is 6/84 (approx. 0.7%) and with 4, it's 36/126 (approx. 20%). 5 squares is the tipping point in a 3x3 where you can have a row and a column both completed which complicates things. I couldn't see a simple mathematical way to calculate how many combinations satisfy the conditions, and checking each one means checking over 100 combinations. Scale that up to the 10x10 grid, and that's many many combinations to check by hand, all for a guess at a number.

for 5 squares you get 84/126 (I think) which is 66.6...% again. for 6 squares you get 72/84 (0.86%) and when you hit 7 it's 100%.

There has to be a better way. A more mathematical way, and a less random way. Any thoughts? I'm looking at pascal's triangle, wondering if there's some answers there.

 

[Atticus837]

At this point I could write a python script to brute force it, but There has to be a more elegant solution. What have I missed?