I have a large probability class and would like to do some memorable demonstrations of Markov chains for them. Does anyone have any recommendations for a "low-tech" demo that doesn't involve computers (or at least, one that minimizes this requirement)? I have access to all kinds of physical props like dice, poker chips, coin tosses, and so forth.
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2 Answers
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Per Wiki:
https://en.wikipedia.org/wiki/Markov_chain#Gambling
Suppose that you start with \$10, and you wager \$1 on an unending, fair, coin toss indefinitely, or until you lose all of your money. If $X_n$ represents the number of dollars you have after $n$ tosses, with $X_0 = 10$, then the sequence $\{X_n: n\in \mathbb{N}\}$ is a Markov process. If I know that you have \$12 now, then it would be expected that with even odds, you will either have \$11 or \$13 after the next toss. This guess is not improved by the added knowledge that you started with \$10, then went up to \$11, down to \$10, up to \$11, and then to \$12.
I recommend to make the following modifications though:
Use a smaller starting amount (3 or 5 or the like) so you can run a few examples. Think it will take a while to exhaust \$10 bankroll.
Flip the coin, cover it, and then call on a student in class to guess it. Pick different students at random. Pick a student (with good handwriting) or use a TA, to graph or tally the experiment as it goes, on the chalkboard. (Makes it more interactive...human factor.)
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The simpliest example I know is the ball game with at least three people: it starts with one person, having the ball, and every second (s)he throws it to another person randomly. Question: what's the probability that the ball gets to person X after n seconds?
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1$\begingroup$ If you do this with humans, it will be highly non-markow. Humans are bad at randomness and will tend to distribute the ball more evenly than random. $\endgroup$– JasperDec 21, 2018 at 12:18