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S2000magician
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Quote:
On 2009-01-16 22:28, Magnus Eisengrim wrote:
It was implicit that the coin was fair, and that the tosses were random.

Why is it implicit that the coin is fair?

I'm sorry, but I don't agree with that. It's a simplifying assumption, but to allow the students to base their answers on simplifying assumptions is dangerous.

The question is (presumably) supposed to test some sort of knowledge or ability. What is this question testing, then? That if a student assumes that the probability is 50% he can conclude that the probability is 50%? Not a particularly useful skill.

Quote:
On 2009-01-16 22:28, Magnus Eisengrim wrote:
I hope you never see Rosencrantz and Guildenstern are Dead; you'd make the opening scene intolerable. Smile

I'd be OK: I don't talk during performances (plays, movies, what have you). I just wish that my younger son had developed that trait! Smile
balducci
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Quote:
On 2009-01-16 20:04, S2000magician wrote:

Mary tosses a coin five times. All five times it lands heads. On the sixth toss, what is the theoretical probability of it landing tails? [Emphasis added.]

He answered: Fifty percent.

She said: You're right!

I said, "That's the stupidest thing I've ever heard!"

First, I have no idea what the word "theoretical" means in that question. To what theory is the author referring? (My guess is that he had no particular theory in mind.) The only one that makes sense to me is a corollary to Bayes' Theorem that states that in the absence of evidence to the contrary, congruent outcomes should be assigned equal probabilities. That would suggest that "heads" and "tails" - congruent outcomes - should each have a probability of 50%.

Something called Bayesian statistics or Bayesian inference is what you are looking for. It uses Bayes' theorem but goes a bit beyond it. You'd begin with a prior assumption of, say, 50/50 for head/tail but use the observed data to update those probabilities. I'm just back from the bar and a little tipsy and I refuse to do the calculations right now, but 5 heads would certainly shift the probabilities to something more in line with your intuition that heads is now more likely than tails.

People (I don't mean you) make many claims about how bad statistics is, but most of the time they just don't understand what modern statistics really is about. Yeah, people can lie with statistics but only dishonest lobbyists and politicians do so. Your typical neighborhood, legitimate, professional statistician would not.
Make America Great Again! - Trump in 2020 ... "We're a capitalistic society. I go into business, I don't make it, I go bankrupt. They're not going to bail me out. I've been on welfare and food stamps. Did anyone help me? No." - Craig T. Nelson, actor.
Magnus Eisengrim
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Balducci, that is precisely his point. I'm suggesting that [it should have been obvious that] the radio guy meant nothing of the kind, that he was using "obvious" assumptions to make the rather simple point that the coin has no memory of the past.

S2000 doesn't seem to agree. Fair enough, I didn't hear the program. The different interpretations of the problem are:

S2000 (and presumably Lobowolf:) Ignoring what you normally believe about coins and though experiments using coins, what would these outcomes tell you about the coin and, hence about future outcomes.

Eisengrim: Accepting what you normally believe about coins, what does the stated outcome tell you about likely future outcomes.

Surely an honourable disagreement.

John
The blood-dimmed tide is loosed, and everywhere
The ceremony of innocence is drowned;
The best lack all conviction, while the worst
Are full of passionate intensity.--Yeats
S2000magician
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Quote:
On 2009-01-16 23:35, balducci wrote:
Something called Bayesian statistics or Bayesian inference is what you are looking for. It uses Bayes' theorem but goes a bit beyond it. You'd begin with a prior assumption of, say, 50/50 for head/tail but use the observed data to update those probabilities. I'm just back from the bar and a little tipsy and I refuse to do the calculations right now, but 5 heads would certainly shift the probabilities to something more in line with your intuition that heads is now more likely than tails.

Given 5 heads and 0 tails the Bayesian estimate of the probability of getting a head is 6/7; the Bayesian estimate of the probability of getting a tail is 1/7.


Quote:
On 2009-01-16 23:43, Magnus Eisengrim wrote:
I'm suggesting that [it should have been obvious that] the radio guy meant nothing of the kind, that he was using "obvious" assumptions to make the rather simple point that the coin has no memory of the past.

If it had been merely "the radio guy" (gal, actually) who posed the question, I wouldn't have cared more than a whit: she, presumably, isn't a mathematician.

I object to having such a poorly-designed question on an exam which will allow someone to graduate (or prevent someone from graduating) from high school. If a student makes the perfectly reasonable assumption that the coin appears to be biased toward heads, he'll get the wrong answer. (And I still have no idea what that word "theoretical" is supposed to mean in that question.)

(I don't know whether this is a multiple-choice test or a fill-in-the-answer test: the woman on the radio didn't offer any answer choices, but that doesn't mean that there weren't any there.)
S2000magician
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Quote:
On 2009-01-16 23:43, Magnus Eisengrim wrote:
I'm suggesting that [it should have been obvious that] the radio guy meant nothing of the kind, that he was using "obvious" assumptions to make the rather simple point that the coin has no memory of the past.

If it had been merely "the radio guy" (gal, actually) who posed the question, I wouldn't have cared more than a whit: she, presumably, isn't a mathematician.

I object to having such a poorly-designed question on an exam which will allow someone to graduate (or prevent someone from graduating) from high school. If a student makes the perfectly reasonable assumption that the coin appears to be biased toward heads, he'll get the wrong answer. (And I still have no idea what that word "theoretical" is supposed to mean in that question.)

(I don't know whether this is a multiple-choice test or a fill-in-the-answer test: the woman on the radio didn't offer any answer choices, but that doesn't mean that there weren't any there.)
Magnus Eisengrim
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S2000, Oh, I misunderstood your point. In that case, I agree completely that such a question should not be on the exam unless either it was open-ended and the markers had some sophistication, or if more information was given.

Somehow, I got the impression that you were annoyed by the mathematics, but your annoyance at the context makes much more sense.


John
The blood-dimmed tide is loosed, and everywhere
The ceremony of innocence is drowned;
The best lack all conviction, while the worst
Are full of passionate intensity.--Yeats
ed rhodes
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Quote:
On 2009-01-16 20:04, S2000magician wrote:
Quote:
On 2009-01-16 18:31, LobowolfXXX wrote:
Levels of sophistication: You're watching a roulette table in Vegas. Someone offers you a hundred bucks if you can correctly pick the next color that will come up. While you've been standing there, the last 10 spins have been red.

Level 1: Bet on black, because it's "due."
Level 2: Flip a coin, because the wheel has no memory, and black and red are
equally likely.
Level 3: Bet on red, because if there's anything at all to go on, it's that the
wheel appears more likely to have a red bias than a black bias.

Amen!

A few years ago I was listening to KPCC (the local NPR station) as I was on my way to work; two people were discussing the exam that high school students in California will (eventually) have to take to graduate. One said that she was going to read one of the questions to see if the other could answer it correctly. She read:

Mary tosses a coin five times. All five times it lands heads. On the sixth toss, what is the theoretical probability of it landing tails? [Emphasis added.]

He answered: Fifty percent.

She said: You're right!

I said, "That's the stupidest thing I've ever heard!"

First, I have no idea what the word "theoretical" means in that question. To what theory is the author referring? (My guess is that he had no particular theory in mind.) The only one that makes sense to me is a corollary to Bayes' Theorem that states that in the absence of evidence to the contrary, congruent outcomes should be assigned equal probabilities. That would suggest that "heads" and "tails" - congruent outcomes - should each have a probability of 50%.

However, note the premise of that corollary: in the absence of evidence to the contrary. That doesn't apply here: there's lots of evidence to the contrary: five heads in a row! (For those of you who think that five heads in a row isn't much evidence, imagine that she'd said "ten times", or "one hundred times", or "ten thousand times"; the specific number five surely isn't the point of the question.)

What's more disturbing is that the answer to the question requires the student to recall some theoretical (there's that word again!) result from a book, while ignoring the evidence that's staring him in the face.

I e-mailed KPCC as soon as I got to work.

They didn't reply.

Sigh.


Unless you can show that the person flipping the coin is affecting it in some way. Then the odds of the coin flip are still 50/50 no matter what the history of the coin flip has been.
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daffydoug
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Quote:
On 2009-01-16 16:27, Doug Higley wrote:
If a thing is One In A Million...that means this 'thing' can happen anywhere between ONE and a MILLION. When ever it does hapen, it changes the odds...if it occurs at 600 then statistics will show it's One In 600 not a million. So of what purpose are the odds...if it happens at #10 is it still one in a million or One in 10?

If it's one in a million that a ladder will fall off a building and hit you on the head...then over 300 intances of this exact thing should be occuring every day in the USA...or is it every minute? So let's say statistics show that if you shoot a BB Gun there is a one in two million chance you'll put your eye out. You figure those are safe odds for now and pick up the Red Ryder. On shot number 27...you put your eye. Oops.

A one in ten BILLION chance can still occur an event #5..or 10,000 or 27 just as readily...so what is the point?

I play a LOT of poker at Full Tilt (play money)...I am consitantly on the Final table (9 out of 90 players) in Tournaments and often make it to the final 2 players. I completely ignore odds and statistics. If a card is out of the deck...there are NO odss that it will show (unless reintroduced) which throws off the odds of all the other cards in the spread as well. I guess I play Full Tilt Chaos? My wife is a 'card counter' and thinks I'm just very lucky.

So...Odds are bogus. Good Faries and Guardian Angels Rule. I don't know from what I'm speaking...it's just an interesting side thought at the moment.


That's why I stick with chess. No luck involved!! Nope. It's all determined by skill, and I lose just about every time. But at least there are no surprises.. Smile
The difficult must become easy, the easy beautiful and the beautiful magical.
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