I have been unable to solve the following exercise from my Probability and Statistics book.

I succeeded in expressing observations (ii) and (iii) as mathematical statements:

(ii)

for

= 1,2,3,4.

(iii)

I also know the probabilities sum to 1, so

To solve the exercise, I need to find

, but I have been unable to do so. I tried rewriting the sum of probabilities like so:

I have tried other approaches, but it seems I am aimlessly solving equations that are just restatements of each other. I believe if I could find

and

, then I could find

and

. If anyone could point me in the right direction, I would appreciate it.

By the way, the solutions manual says the answer is (C) 0.27.