For several years now the State of California has been threatening to require high school seniors to pass an exit exam before granting them their diplomas. This requirement was to have gone into effect a few years ago, but sees delay after delay as various parties argue whether it's fair or biased or whatever.

Two-and-a-half years ago, as I was driving to work, I heard this question on the radio - KPCC, a local NPR station; it's a question from the exit exam. A couple of days ago I heard another report on the exam saying that it was going to be implemented in 2006 (yeah, right!); that report brought this question to mind.

Mary tosses a coin five times. All five times it comes up "heads". What is the theoretical probability of getting a "head" on the sixth toss?

I'll post my thoughts on it after some discussion.

 

voted 50% assuming it's a "fair coin"

otherwise, if we can't assume that it's a fair coin, then the probability is greater than 50%

 

What does it mean to be a "fair" coin?

 

Not this one:

http://cgi.ebay.com/Double-Two-2-Sided-Hea...1QQcmdZViewItem

 

that each flip has a 50/50 chance of landing heads or tails, if that is true, then each flip is an independent event whose outcome has nothing to do with any other flip

 

How many sequential "heads" results would you need to see before you started doubting the validity of that assumption?

 

While it is true that if each flip has a 50% chance of landing "heads" and a 50% chance of landing "tails" then all flips are independent, the same is true if each flip has a 70% chance of landing "heads" and a 30% chance of landing "tails"; i.e., while fairness implies independence, independence doesn't imply fairness.

I asked what you meant by a "fair" coin for a particular reason. You said, in essence, that you assume that the probability of getting "heads" is 50%. In that case, the original question (about Mary's coin) boils down to this: If you assume that the probability of getting a "head" is 50%, what is the probability of getting a "head"? In that case, what is the point of the question (in an exit exam)?

 

People would still get it wrong.

 

Are you being asked this question in the beginning of the test or after the fifth coin toss?

 

Ok, my answer was poorly worded. A fair coin has a 50/50 chance of landing either heads or tails on any given flip. That's all a fair coin means to me.

I'm not saying that fairness implies independence. The two are separate. I have assumed that each coin flip is an independent event, period. So, if your coin has a 70% chance of landing heads and 30% chance of landing tails, then that's what the probabilities will be for every single flip, regardless of what happened on the preceding flip. This is assuming that all variables (i.e., wind, how the person flips the coin, etc.) remain the same from flip to flip.

Flip. Just thought I'd throw that in there for good measure. Ok, one more. Flip.

If it's a fair coin, all variables remain the same over each flip (there's that word again), and the probability of getting a head is 50%, then my answer is there's a 50% chance of getting a head on the sixth flip. Or the 500th flip, for that matter. It doesn't make any difference.

What's the point of the question? I don't claim to know. Having given it several seconds of thought so far, I'd say it's to see if students can employ logic when analyzing trends and patterns. Enlighten us on the point of the question.

By the way, you posted this same question about 2 years ago in this very forum.