These are some rather odd questions that I got presented with at my university today. I'd really like to get them solved and I know just about what I need to solve them, just not sure how to go about putting it into practice.

The first has to do with probabilities and infinite series, I know that much.

I tried taking the probabilities:A board game consists of four positions labeled A, B, C, and D. Whenever you reach position A or B, you roll some dice whose results determine what position you will move to next. Ah, but these dice are loaded.

Fromposition A, there is a 1/4 chance ofmoving toposition B, a 1/12 chance ofmoving toposition C, and a 1/6 chance ofmoving toD.

Fromposition B, there is a 1/3 chance ofstayingat B, there is a 1/3 chance ofmoving toposition A, a 1/6 chance ofmoving toposition C, and a 1/6 chance ofmoving toD.

Whenever you reach position C or D the game is over and you win some cake.

Suppose you begin the game at position A, what is the probability you end the game at position C

P(Moving from A to C):1/12

P(Moving from A to B to C):1/4 x 1/6 = 1/24Here is where I get stuck. Normally I would just keep drawing out the probabilities to the situation I want. Not sure how to do it since, you could possibly move back to either A or stay at B after B. If anyone has any tips about how I can go about uncovering the correct infinite sequence for this I would be much obliged.

Also it is a strange question but I got this question too:

The zero functions f(x) = 0, and g(x) = 0, while correct in this context are not the desired answers.Find two functions f(x) and g(x) such that the integration of the product (f(x) x g(x)) is equal to the product of the integration of f(x) and the integration of g(x).

Integration(f(x) x g(x)) = Integration(f(x)) x Integration(g(x))

This seems like an easy questions but the more I got to thinking about it, I could not off the top of my head come up with two functions for which this is true.