A marker is placed at zero on the number line and a fair coin is flipped. On each flip we move one unit to the right. If it lands on heads, the marker is moved one unit up. If it lands on tails, the marker is moved one unit down.
If the first flip is a head the marker goes to position A(1,1). If the second flip is a head the marker goes to position B(2,2). If the third flip is a tail the marker goes to C(3,1). So with each flip we move one unit to the right on the x axis, and +/- one unit on the y axis depending on the landing on the coin.
When the marker is stationed at 0, I know that it has a 50-50 chance of going to either 1 or -1. So in this simple case the probability is proportional to the distance traveled.
Do I still get to say the same thing about the probability of the marker going to 1 in raport to the probability of it going to -2? Can I now say that the marker has a 66,6% chance of going to 1 and a 33,3% chance of going to -2? (Remember that initially the marker is stationed at 0 and it can reach any of the two points in every possibile way. There are no conditions regarding how it gets there or how long it takes to get there)
What about 1 and -3? Can I say that when the marker is stationed at 0 it has a 25% chance of going to -3 and a 75% chance of going to 1?
In other words, just like the title says, are these probabilities proportional to the distance traveled in a random walk? If the raport between the distances is 3:1 is it correct to say that the probabilities have a raport of 1:3?
The second part of my question refers to the initial position of the marker. What if all the other initial conditions stay true, but the marker is initially placed at 0.2? I now know that I have a 50% chance of going to 1.2 and a 50% chance of going to -0,8. But what are the chances of getting to -1 and what are the chances of getting to 1? How does this change in initial position affects things? Do probabilities remain proportional to the distances? Since there is a raport of 1.2/0,8 between the two distances, can I conclude that the marker has 60% chances of getting to 1 and 40% chances of getting to -1? (60%/40% = 1,2/0,8)