[QUOTE]Originally posted by magician

 

[QUOTE]Originally posted by magician
Three questions:

1.

 

"The actual, correct explanation", huh?

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.

 

You're absolutely right. And the limited evidence that you have so far is that it's steering away from 50-50 and toward a strong bias for heads. In my opinion, that's the single most important observation that high school students should be encouraged to formulate. And it's exactly the one their answer crushes underfoot.

 

If the original question had had Mary getting 10 heads in a row, would your answer be different? How about 20 in a row? Fifty? One hundred? One thousand? One million?

According to their answer, you should treat all of these situations the same. They might allow you to say "Wow!" when the number exceeds 20, and to say it louder as the number goes up, without penalizing you.

 

I just meant your explanation, since you understand this stuff better than myself. I've only taken a few mandatory statistics courses, and honestly have never bothered trying to apply any of it in real life.

Anyway, in this case, I think it's just an ambiguously phrased question. The purpose of the question, IMO, is trying to test one of the following...

#1.
Whether students realize that the probability of getting heads/tails for a fair coin on a single flip is 50%, regardless of previous results. This would show an understanding that each flip is an independent event.

I do believe defining the coin as fair would clarify things, since that describes some important and relevant attributes of the coin.

#2.
Whether students can think critically and see that, with only the given information, and without making any inferences, the probability of getting heads on the 6th flip is higher than the probability of getting tails.

Given that this is a high school exit exam, my guess is #1. On many standardized tests, I feel that it's often against the test-taker's interests to think over any of the problems too thoroughly.

However, I wholely agree with your explanation. If we know it's a fair coin, then the probability of either outcome is 50% for any toss, period. If we go based only on what the question states, then the 6th toss is more likely to produce heads.

 

[QUOTE]Originally posted by cyber_x
Anyway, in this case, I think it's just an ambiguously phrased question.

 

I'm reminded of some advice given to students in veterinary school: If the dog that is lying cut-open on your operating table doesn't look like the picture that the author put in your textbook, the dog is right.

 

magician-- have you seen that new television show, "Deal or No Deal"?

 

Theoretical probability the person who named the segment "Are It True" actually paid attention in English class...ZERO!!!!

It should be "IS It True" or "ARE THEY True".

How can they do math when they can't even use correct English?

Warren