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Message: <blockquote data-quote="orsal" data-source="post: 1737508" data-attributes="member: 16016">Here's an exercise I occasionally give my intro prob/stats students: I give them three sequences of H's and T's, all the same length (at least 60 tosses), and ask them which of the three was a simulated coin toss. In acutality, one of them is a simulated toss of a fair coin (independent probabilities, 0.5 for each), one is a simulated toss of a weighted coin (independent, but tails significantly more likely than heads), and the third is cooked so that heads will more often be followed by tails, and vice versa. (I'll suppose they're given in that order, although I actually change the order from time to time.)The first thing nearly everyone looks for is the total number of heads and tails. They (correctly) expect close to half of each, and on that basis rule out sequence #2. It's harder to choose between #1 and #3, but the most common criterion used is long homogeneous strings. A lot of students will look at #1 and decide that there's a string of consecutive H's or consecutive T's that's suspiciously long. #3, of course, is cooked up to avoid long homogeneous strings, and to most people that "feels" more random. It's a surprise to learn that randomness generally does produce patterns of all kinds writ small. Toss a coin enough times, and you will at some point get 289 consecutive heads. A string of 7 consecutive H's or 7 consecutive T's happens one time in 64; if you toss 70 coins, there are 64 strings of 7 consecutive tosses, and on average one of them will be homogeneous. It's more complicated to calculate the actual probability of getting at least one homogeneous string, but it's better than even.So the very feature that should suggest that #3 is not truly random, is what leads most people to guess it is. The situation would get even worse if I asked you to write down a pseudo-random string of H's and T's off the top of your head. (Well, maybe not if I asked you. Maybe you're weird. But if I asked almost anybody.) I'd likely never see more than three consecutive H's or T's. People's attempts to imitate randomness invariably err by producing too much alternation.Likewise, if you tossed a d6 100 times, chances are you'd get 2-3 homogeneous runs of 3 (of anything), since the probability that a particular run of three tosses will be homogeneous is 1/36, there are 98 runs of three, and 98/36=2.7 But if I asked someone (who hasn't read this post or anything similar) to write down 100 numbers from 1-6 "as if they came from a die", I'd bet I'd not see a single 3-in-a-row.Note that if you roll a fair d6 100 times, any *specific* result is as likely as any other. But there are some *patterns* that suggest non-randomness, and studiously avoiding patterns such as homogeneous streaks leads to a different kind of pattern which is just as improbable.</blockquote>

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