I put 4 cups in a line. In each cup I place 1 red ball and 1 blue ball.

I take a ball at random from the first cup, and put it in the second. Then I take a ball at random from the second cup, and place it in the third. Then I take a ball at random from the third cup, and place it in the fourth. Then I take a random ball from the fourth cup.

(a) What is the probability that this fourth ball is red?

(b) What is the probability that this fourth ball is red, given that the first ball is red?

Now I set up 7 cups in the same way, and perform the same experiment.

(c) What is the probability that the 7th ball is red, given that the first ball is red?

Can you form a general conjecture based on your answers to (b) and (c)?