A bag contains 4 red marbles and 4 blue marbles. Leanne and Sylvia play a game where they draw four marbles in total, one by one, uniformly at random, without replacement. Leanne wins if there are more red than blue marbles, and Sylvia wins if there are more blue than red marbles. If there are an equal number of marbles, the game is tied.

(a) Let A1 be the event that the first marble is red and let A2 be the event that the second marble is red. Are A1 and A2 independent?

(b) What is the probability that Leanne wins the game?

(c) Given that Leanne wins the game, what is the probability that all of the marbles were red?

Now, suppose the bag contains 8 red marbles and 4 blue marbles. Moreover, if there are an equal number of red and blue marbles among the four drawn, Leanne wins if the third marble is red, and Sylvia wins if the third marble is blue.

(d) What is the probability that the third marble is red?

(e) Given that there are k red marbles among the four drawn, where 0 k 4, what is the probability that the third marble is red? Answer in terms of k.

(f) Given that the third marble is red, what is the probability that Leanne wins the game?