View Poll Results: What is the theoretical probability of getting a "head" on the sixth toss?
Probably less than 50%, but we can't be certain
0
0%
Voters: 25. You may not vote on this poll
One More Probability Problem
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From: Newtown,PA
To me this question seems unfair. There is no way of telling what the grader is looking for. Like stated above, judging by the odds, it appears to be a "trick" coin. On the other hand, maybe the grader is looking for the "logical" response of 50/50. This would be a case where, not enough information ,should be a option.
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From: SF Bay Area
magician's explanation the last time around:
Commentator K asked this question of commentator J.
Commentator J replied, "Fifty per cent."
Comemtator K stated, "You're absolutely right!"
Listener B--yours truly--said "That's the stupidest thing I've ever heard!" and sent a scathing e-mail to NPE when he got to work that morning.
No reply.
The inclusion of that word "theoretical" in the question is very bothersome, because I have no idea what the author means: To which theory is he referring? If, as cyber_x suggests, he's referring to Bayes' Theorem (and I believe he is, whether he knows it or not), he's trying to apply the corollary that says that "in the absence of evidence to the contrary, congruent outcomes should be assigned equal probabilities." I fully agree both with the theoretical result and the application of that result to this problem.
Unfortunately for the author, the application of this result only occurs in the negative. The hypothesis is, "in the absence of evidence to the contrary". In this problem there is a ton of evidence to the contrary--five head in a row, to be exact. This is a very improbable outcome for a fair coin, a certainty for a two-headed coin, and a very likely outcome for a coin heavily biased toward heads.
What I resent in particular is that the effect of this answer will be to further condition high school students to rely on their preconceived notions about how things ought to behave and to ignore solid contradictory evidence that is staring them in the face. I believe that this is an extremely dangerous practice. (I'm quite serious here.)
Suppose you asked the question slightly differently: if the coin came up heads five times in a row, on which side should you place an even-money bet for the sixth toss: heads or tails?
Unfortunately, you will find a large segment of the population who will say "tails" and cite some mythical "law of averages" to justify their answer. The correct answer is "heads". If the coin is fair, it makes no difference on which outcome you bet. If the coin is unfair, the evidence is that it is biased toward heads.
Please don't suggest that the author could have improved the question by saying, "Mary tosses a fair coin five times . . . ." That gets us back to the stupid question, "If you assume the probability is 50%, what's the probability?"
By the way, if you apply Bayesian statistics to the outcome of the first five tosses, you get an estimate of the probability of getting a tail on the sixth toss of 1/7. Bayesian analysis forms the basis of a very successful method I have developed for betting on football games, and that I recently have adapted to baseball, basketball, and ice hockey.
Commentator J replied, "Fifty per cent."
Comemtator K stated, "You're absolutely right!"
Listener B--yours truly--said "That's the stupidest thing I've ever heard!" and sent a scathing e-mail to NPE when he got to work that morning.
No reply.
The inclusion of that word "theoretical" in the question is very bothersome, because I have no idea what the author means: To which theory is he referring? If, as cyber_x suggests, he's referring to Bayes' Theorem (and I believe he is, whether he knows it or not), he's trying to apply the corollary that says that "in the absence of evidence to the contrary, congruent outcomes should be assigned equal probabilities." I fully agree both with the theoretical result and the application of that result to this problem.
Unfortunately for the author, the application of this result only occurs in the negative. The hypothesis is, "in the absence of evidence to the contrary". In this problem there is a ton of evidence to the contrary--five head in a row, to be exact. This is a very improbable outcome for a fair coin, a certainty for a two-headed coin, and a very likely outcome for a coin heavily biased toward heads.
What I resent in particular is that the effect of this answer will be to further condition high school students to rely on their preconceived notions about how things ought to behave and to ignore solid contradictory evidence that is staring them in the face. I believe that this is an extremely dangerous practice. (I'm quite serious here.)
Suppose you asked the question slightly differently: if the coin came up heads five times in a row, on which side should you place an even-money bet for the sixth toss: heads or tails?
Unfortunately, you will find a large segment of the population who will say "tails" and cite some mythical "law of averages" to justify their answer. The correct answer is "heads". If the coin is fair, it makes no difference on which outcome you bet. If the coin is unfair, the evidence is that it is biased toward heads.
Please don't suggest that the author could have improved the question by saying, "Mary tosses a fair coin five times . . . ." That gets us back to the stupid question, "If you assume the probability is 50%, what's the probability?"
By the way, if you apply Bayesian statistics to the outcome of the first five tosses, you get an estimate of the probability of getting a tail on the sixth toss of 1/7. Bayesian analysis forms the basis of a very successful method I have developed for betting on football games, and that I recently have adapted to baseball, basketball, and ice hockey.
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From: SF Bay Area
I agree with this explanation if we are to assume only the information given in the question. But, if you ask me, that is one poorly-bounded question, especially if it were to be administered to high school students.
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From: SF Bay Area
Originally Posted by Cyclon36,Oct 19 2005, 08:25 PM
what do you suppose the probability is of getting head from mary? 
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From: Nowhere
Originally Posted by Cyclon36,Oct 19 2005, 03:25 PM
what do you suppose the probability is of getting head from mary?
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That is the problem with most "standardized tests" The questions are poorly worded. Some people would understand that if a coin gave heads five times in a row, it was weighted to fall on the heads side, or maybe both sides were heads. But probability says a coin, assuming it is a standard American coin has a 50% chance of landing heads up each time independently of previous tosses.
So the question is really is the tester trying to trick me into a wrong answer or is the tester so stupid they worded it incorrectly? The answer is, when taking standardized tests, pretend you are the test writer. Who are the target audience? What should the target audience know? What was the tester trying to ask? On a highschool exit exam, the problem was probably a poorly worded question and the "correct answer" is 50%.
If the exam was a GRE, it probably a trick question and the "correct answer" is more than 50%
Also these are timed tests. Do not get all twisted up over one question. Just guess quickly and go on.
As society depends more and more on standardized tests, the validity and quality control on the test will become more important. This first generation of kids that have to pass these highschool exit exams are at a terrible disadvantage because these first tests have some real weaknesses.
So the question is really is the tester trying to trick me into a wrong answer or is the tester so stupid they worded it incorrectly? The answer is, when taking standardized tests, pretend you are the test writer. Who are the target audience? What should the target audience know? What was the tester trying to ask? On a highschool exit exam, the problem was probably a poorly worded question and the "correct answer" is 50%.
If the exam was a GRE, it probably a trick question and the "correct answer" is more than 50%
Also these are timed tests. Do not get all twisted up over one question. Just guess quickly and go on.
As society depends more and more on standardized tests, the validity and quality control on the test will become more important. This first generation of kids that have to pass these highschool exit exams are at a terrible disadvantage because these first tests have some real weaknesses.
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From: Longmont, CO
A dumb person would say there's a low possibility of getting another "head" because so many already showed up.
A smarter person would say it's a 50/50 chance, because previous results do not influence future ones. It's not like the coin remembers its last flip.
A truly intelligent person would make no assumptions and ask "Is it a two-headed coin?"
A smarter person would say it's a 50/50 chance, because previous results do not influence future ones. It's not like the coin remembers its last flip.
A truly intelligent person would make no assumptions and ask "Is it a two-headed coin?"
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From: Yorba Linda, CA
Originally Posted by CARNUTMAMA,Oct 19 2005, 12:36 PM
But probability says a coin, assuming it is a standard American coin has a 50% chance of landing heads up each time independently of previous tosses.
"Probability" doesn't say a blessed thing about whether a standard American coin (whatever that is: twenty-dollar gold piece, Walking Liberty half, buffalo (bison, actually) nickel, . . . .) has a 50% chance of landing heads up, or a 49% chance, or a 51.239627% chance, or some other chance. Probability theory simply gives us tools to analyze the consequences of the assumption that a standard American coin (whatever that is) has a 50% chance of landing heads up, and to estimate what that chance is.