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A walker takes 10 steps, each one to the left or one to the right with the same chance…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A walker takes 10 steps, each one to the left or one to the right with the same chance. What is the probability that the walker is back at the start? Estimate it with 100 000 walkers, and compare it with the exact answer: the number of ways to choose 5 right steps out of 10 steps, divided by the number of all paths, .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Plan. One row for each walker and 10 columns of steps. The final position is the sum of a row. The walker is back at the start if the sum is zero.
Check. , and , so the exact answer is . The simulation gives 0.2465.
Answer: about 0.246: nearly one walker in four is back at the start after 10 steps.
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