# What's the probability a random game of Connect 4 ends in a draw?

On a standard Connect-$$4$$ board ($$7$$ columns & $$6$$ rows), if two players take turns making moves by selecting an available column uniformly at random, what is the probability the game will end in a draw (i.e. all $$42$$ spots are filled and neither player has $$4$$-in-a-row anywhere in the board)?

I believe the probability of a draw is highly unlikely ($$<1$$%). There are just too many places on the board for $$4$$-in-a-row to occur. I've simulated a few games and none have made it to a draw.

Here is my way of getting a rough estimate. There are $$69$$ locations on the board to get $$4$$-in-a-row (counting horizontal, vertical, and diagonal runs of 4 cells). For each, the probability of all $$4$$ the same color is $$\frac{1}{8}$$ since the first piece can be either color, and the remaining $$3$$ each have probability $$\frac{1}{2}$$ to match the first. Thus there is a $$\frac{7}{8}$$ probability the four pieces are not the same color.

For the whole board, the probability of a draw is found by $$(\frac{7}{8})^{69}\approx0.01$$%

I know this isn't the exact answer, because I didn't consider factors such as there need to be an equal amount of both colors, the players take turns, each cell is part of multiple runs of $$4$$-in-a-row, etc. But I expect my estimate is pretty good. My question is how to calculate the exact probability of a draw, and also the probabilities of player $$1$$ and player $$2$$ winning (I expect going first gives player $$1$$ an advantage over player $$2$$)?

I know the total number of Connect-$$4$$ games can be found on OEIS and elsewhere online, but it isn't clear how many of those $$4.5$$ trillion games end in a draw.

• Simulated a "few" games? How many is a "few"? Mar 5 '20 at 6:50
• A couple hundred, though my programming skills are lacking. I just randomly populated a 6 × 7 grid in excel with 1s and 0s (21 of each) and looked for 4-in-a-row. There are problems with that method (e.g. arrangements that are impossible in regulation play can come up), but I think it's a fair way of seeing if my estimate is somewhat accurate. Mar 6 '20 at 5:21