How do you explain the math solution from an intuitive point of view?

I wish I could remember where I heard this problem. I worked out the solution, but I still can’t explain at an intuitive level why it should work out the way that it does. Here is the problem.

You are at a casino playing the roulette wheel. Someone comes over to you and complains about the house having the advantage. He suggests that you make a side bet with him and keep the money between the two of you. He proposes that you each choose a permutation of length 3 of the colors red and black and the winner will be the one whose choice comes up first on consecutive spins of the roulette wheel. To make things fair, you will be allowed to choose first.

I trust that if this happened to you, you would sense, apart from any mathematical odds, that you were being hustled. Why is this complete stranger making this offer? You would be absolutely right, but on the face of it, if seems reasonable. If there were eight of you, each choosing one of the possible orderings of 3 choices of red and black, one of you would win in 3 spins with equal likelihood.

Consider the most extreme case. Suppose you had chosen red, red, red. What would the other person have chosen? If you do not win in the first 3 spins (ignoring 0 and 00), you are guaranteed to eventually lose. Then I would appreciate it if you could explain to me in simple terms why, for every choice, there is another that is more likely to occur first.