[Physics] Feynman random walk

probability

In Richard Feynman's lectures on physics, chapter six, part 3, he explains something called the random walk, in which, in a succession of trials, a system moves forward one step or backward by one step, each with probability one half.

He comes to the conclusion that the most likely result is that there are more forward steps than backward steps or backward steps than forward steps.

For me, this is counterintuitive.
If heads on a coin represents a forward step, and tails represents a backward step, surely the most likely result is that there will be the same amount of forward steps and backward steps?

Best Answer

You are mixing here two kinds of most likely. First, you are correct, that the same amount of forward and backward steps will be the most probable of all possible outcomes. Second, if you add probabilities of all outcomes with unequal result, you will see that getting some unequal result will be much more likely.

The outcomes of coin tossing are described by Binomial distribution. To give a perspective, the probability of getting exactly $500$ heads out of $1000$ fair coin tosses is $0.025$, and this is the most likely outcome (among a thousand of others!)

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