In one of my homework problems, I'm struggling with the phrase: "with probabilities inversely proportional to the distances to the corner from the current position".
The three distances are 3 cm, 4 cm, and 5 cm, and I have to assign probabilities to these three events totaling 1 in accordance with this inverse proportionality rule.
I constructed the following equation:
(1/3)x+(1/4)x+(1/5)x=1
Then, I solved for x and got x=(60/47), and multiplied this with the (1/3), (1/4) and (1/5). My rationale was to find a constant factor so that I could assign probabilities to these three events totaling 1, but I do not think this is correct.
You are almost done.
The first row of the transition matrix should tell you the probabilities for going from corner $1$ to $2$, $3$ and $4$. Those must add to $1$ and should be in proportion $1/4:1/5:1/3$, the reciprocals of the distances. Since $$ \frac{1}{4} + \frac{1}{5} + \frac{1}{3} = \frac{47}{60} $$ (your arithmetic is correct) you have to scale the fractions by $60/47$. Then they will sum to $1$ and so represent probabilities. So the first row of your transition matrix will be $$ 0, \frac{15}{47} , \frac{12}{47} , \frac{20}{47} $$
(I wish the author of the problem had labeled the vertices $A,B,C,D$ rather than $1,2,3,4$.)