Warm-up: Symmetry for the win! <3

Probability Level 4

You are on the coordinate grid at the point ( 1 , 1 ) (1, 1 ) and you do a 2 2 -dimensional completely random walk. Remember that each step size is ‘infinitesimal‘ and you can go in any direction, not just the 4 4 cardinal directions. Eventually you will cross the x-axis. What is the probability (to the 5 5 th decimal) that you cross the negative x-axis versus the positive x-axis first? Note: You may also consider a walk with fixed step size and regard the probability as step size approaches 0 0 ; If you prefer this way of thinking about the problem!

Note/Hint : Solvable by symmetry only; no calculator needed!

Note : Here is part 2 2 of the problem!


The answer is 0.25.

Simon Kaib by Simon Kaib , 19, Germany

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1 solution

Simon Kaib
Jul 20, 2019

You have probability 1 2 \frac{1}{2} for crossing the y-axis before crossing the positive x-axis (since they have the same distance from the starting point). Once you are on the y-axis you have probability 1 2 \frac{1}{2} for crossing the negative x-axis before the positive x-axis (again, because they are symmetrical to the y-axis)! Thus you have a probability of 1 2 ⋅ 1 2 = 0.25 \frac{1}{2}\cdot\frac{1}{2}=0.25 for crossing the negative x-axis before the positive x-axis!

8 Helpful 2 Interesting 6 Brilliant 0 Confused

Draw the the 2 lines y=1 and the connection from (0,0) to (1,1). The angles created are 45 and 135 degrees respectively. The probability of first crossing x on "negative side" is 45/180=1/4

Eric Scholz - 1 year, 10 months ago

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