Probability & Statistics/Useful equations

Question
G'day!

this is a strange one but;

I was wondering if you knew of, or have heard of any entertaining uses of probability & statistics

i.e. 'if you just guess every answer on a four question multiple choice test of 30 questions, there is a 1 in 1, 52, 921, 504, 606, 846, 976 chance of getting all correct and but to get all incorrect is only 1 in 5600'

Hugh

Well, I'm not sure how to know if I'm giving the right answer here, but I'll give this a shot. I have two interesting anecdotes/scenarios that you might like.

First, imagine there are 25 people in a room, and that you know (for some reason) how many hairs each one has on his or her head. There is a way to form two groups from these people so that the total number of head-hairs in each group is the same.

What's most surprising about this one is that it's not really probability or statistics, it's something that is always true (not just likely to be true, but 100% certain to be true).

The reason is quite simple -- among 25 people, there are 2^25 ways to select a group. But each group is assigned a number between 0 and 25*200,000 (where 200,000 is a very high estimate on the maximum number of hairs on a head). So if you consider all possible groups, at least two of these groups will have the same number of hairs.

Second, an anecdote which is "folklore" in the sense that it's not clear if it happened and it's told and retold so many times that who knows the original story for sure? Anyway, an old professor in a basic statistics class spent his first day giving some preliminary course information, telling students about the course policies (including academic honest and all that), and then gave them just a little information about probability. Random events occur, and some events are independent - like flipping a coin, and then flipping it again, these produce independent random outcomes. Blah blah blah, nobody pays attention on the first day, so the homework was very simple -- make a list of 100 coin flips at home by flipping a coin, write them down, and hand it in.

The next class, he took up the papers and gave a normal lecture. But on the following class period, he was very angry with the class. He accused them all of cheating, that they all were clearly so lazy they won't even flip coins and instead, they must have simply sat down and written HTHTHHTHTHTHT on their own without flipping a coin. That is surely a very lazy thing to do, considering flipping a coin is not much harder at all and requires no work, no thinking, and no studying.

But the students objected!! Surely they didn't all cheat. How could he punish them all? How does he know they were dishonest? The policies - they all insisted they were aware of them - don't let instructors make such baseless accusations!

And so they learned their first lesson in probability. You see, out of the 200 students in the class, in the 20,000 coin flips, there was not a single run of four-in-a-row the same. As it turns out, the probability of getting no runs of four-in-a-row, given n coin flips, is

F[4](2+n) / 2^n

where F[4](2+n) is the (2+n)th 4-step Fibonacci number. See here for that.

Surely, each student had a 2.3% chance of not having such a run, because that is the value of this expression for n=200. But the probability that not a single one of them had a run of four? Well, that would be 0.023^200, which is a number so astronomically huge that the professor was well within reason when he said "as surely as the sun does not instantly explode for no reason and kill us all, I am indeed sure that you all cheated on this assignment."

Now, the lesson here is obviously not to cheat, but also that what "looks" random, that someone sitting down to write down a "random" sequence of H and T, this is not necessarily what we think it is.

It's a good anecdote anyway

Probability & Statistics

Volunteer

Clyde Oliver

Expertise

I can answer all questions up to, and including, graduate level mathematics. I do not have expertise in statistics (I can answer questions about the mathematical foundations of statistics). I am very much proficient in probability. I am not inclined to answer questions that appear to be homework, nor questions that are not meaningful or advanced in any way.

Experience

I am a PhD educated mathematician working in research at a major university.

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AMS

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Various research journals of mathematics. Various talks & presentations (some short, some long), about either interesting classical material or about research work.

Education/Credentials
BA mathematics & physics, PhD mathematics from a top 20 US school.

Awards and Honors
Various honors related to grades, various fellowships & scholarships, awards for contributions to mathematics and education at my schools, etc.

Past/Present Clients
In the past, and as my career progresses, I have worked and continue to work as an educator and mentor to students of varying age levels, skill levels, and educational levels.