If you describe the scenario this way, then clearly, the chance of flipping tails on the 6th flip is higher given the results of the 5 preceding flips. Hell, who knows if that coin even has a tails side. That wasn't explicitly stated, either.

The issue is that most people, be they high school students or not, would assume -- I am not familiar enough with coins to say correctly or incorrectly -- that the coin is a fair coin which has an equal chance of landing on heads or tails for any given flip. I can recall arithmetic word problems in high school predicated on this very assumption, so it's not entirely the test taker's fault for assuming so. If the assumption is unreasonable, then a large part of the reason is that students have been conditioned to assume it during their studies.

 

A lot less after marriage.

 

My thinking is this. In my mind, "fairness" means that the coin has a 50/50 chance of landing on either heads or tails. "Independence" means that the probabilities associated with the potential outcomes of each flip have nothing to do with the outcomes of previous flips.

But simply having a fair coin doesn't mean that the outcome of each flip is independent of all other flips. What if the person flipping the coin decides to flip it differently because of the outcomes he's observed? Perhaps, instead of beginning the flip with the tails side down, he will begin with the tails side up. Is this nitpicking? Sure, of course. But, in my book, so is questioning someone who assumes that the coin is a fair coin (as we've defined it in this thread).

If we are to assume that the coin is flipped in an identical manner each and every time, then I see what you're saying. It all depends on what assumptions we make.

 

I'm quite certain that this is exactly what the author of the question expects the students to do.

As I wrote in the analysis that cyber_x quoted, that's a horrible expectation on the part of the author. Not horrible because it is unlikely to be satisfied; horrible precisely because it is likely to be satisfied: Here, students, spit out some factoid that you memorized while ignoring all of the evidence staring you in the face. Hardly strikes me as the hallmark of an accomplished educational system.

I'd really like to talk to the author of that question to find out what ability he's trying to test with that question, and, more specifically, what the heck he means by "theoretical".

 

Well I can agree with that to some extent. But I wouldn't single out this one question. Poorly-written exam questions abound, and this particular question is preceded by years of conditioning.

We see coin flips being used to determine possession in sports games. We read word problems involving coin flips in junior high and high school and are told to assume that it's a 50/50 chance between heads and tails. And so on. We're repeatedly conditioned to think that heads and tails both have an equal shot at landing face-up.

I wouldn't place the blame wholly on the author. In all likelihood, he is also a product of this sort of conditioning. There's a lot of crap in the world that doesn't make sense, and I'm personally not that bothered by this particular instance.

 

If the probability is 50/50 on each toss then what difference does it make whether Mary starts with the coin heads up or tails up or uses her left hand instead of her right or lets the coin bounce off her . . . notebook, say, or whatever. The assumption that it is 50/50 on each toss renders all those variables moot.

The definition of independence is P(A & B) = P(A) * P(B).

Because P(A & B) = P(A | B) * P(B) [P(A | B) means the probability of A, given that B has occurred],

P(A | B) * P(B) = P(A) * P(B), or

P(A | B) = P(A).

If the probability 50/50 on each toss, then P(H | anything) = 0.5 = P(H). Hence, the tosses are independent.

 

The difference is how the system is defined. I have defined the system for a toss and for independence differently. And you're right, that's probably incorrect.

For each toss, I thought of the system as consisting of only the coin. Somehow, this coin magically becomes airborne. It now has a 50/50 chance of landing on either of the two sides.

For independence, I thought of the system as consisting of the coin as well as the person flipping it. And the "person flipping it" may undergo changes from flip to flip.

So, the fallacy here is that the two systems don't share the same definition. I am now defeated. Pardon me while I go smoke some weed.

 

You're right, no doubt.

It's that word "theoretical" that I find most bothersome.

 

Theoretically, of course.

 

Actually, it's pretty much proven fact by now.