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Here's a microlesson on a neat subject—random walks!

A chip, starting in position 1, randomly jumps to a reachable neighboring spot and keeps doing so. What's the average no. of jumps the chip will make before it gets to position 6? #math
The key thing to figuring this out is realizing that where the chip goes next ONLY depends on the place it currently is.

Where it was before, in the previous jump or any before it, doesn't determine anything about where it goes next.

(The math lingo for this is "Markovian".)
As a result, when the chip moves, it's as if it were starting the "game" again, except at a different starting position & with an extra jump on the counter!

That means that we can think about the average jumps needed to get to 6 from any other position on the board—2, 3, etc.
Now the trick is to try to relate the average number of jumps to 6 from every starting position.

For example—if I start on 5, I have a 50% chance of reaching 6 in one jump, and a 50% chance of "restarting" the game on 2 with an extra jump.

We can use that to relate avg. jumps!
Intuitively, I can then say that the average number of jumps to get from 5 to 6 is a "50/50 mix" of just 1, and 1 + the number of avg. steps from 2 to 6. So weighing each option by a half and adding them up should get me what I want!

(It seems sloppy, but the math works out.)
If I do this for every starting position, I get a system of equations that relates the average number of jumps to get to 6 from every starting position. This system is now solvable with old-fashioned algebra!

Can you find the expected number of jumps to get from 1 to 6 now?
SPOILER:

The answer is 19 jumps. Seems big, no?

Anyways, systems like these—where something is moving around randomly while following a set of rules—are called "random walks", and this funny technique (called first-step analysis) is just one tool we can use to understand them.
Random walks come in all shapes & sizes, and not all of them have the nice properties that this one does; but that's a story for another microlesson.

If you like, share an example of random walk-like phenomena you know about, or something that you suspect might be a random walk!
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