Chance of Game Ending in 5 trials: Exam Question Analysis

[mynameisfunk]

Ok, This is not a homework or textbook problem; this is a question from an exam I recently took that I would like opinion on the wording of the problem.

Question: A fair coin is flipped until a head is observed and then a fair die is rolled until a 6 appears. Each flip and each roll is counted as a "trial". What is the chance that the game ends in 5 trials?

I put the culprit in bold. I interpreted the definition of a trial to be flipping a coin until you get a head and then you get to roll the die and that is the end of a trial, if you did not get a 6, then you start flipping the coin again. The reason I understood this was because of the word "and", if it had said "Each flip or each roll is counted as a trial" then I believe it would have meant the easier interpretation of just counting the ways you can flip a coin until you get a head and then you start rolling the die and never pick the coin back up.

I solved it as I interpreted it, which I may or may not be wrong, but I took the complement of the event of flipping a head and rolling a 6 which should be 1-[(1/2)(1/6)]=(11/12) then I simply said the chance of it ending exactly on the 5th trial was [(11/12)^4]*(1/12)

Sorry for the lack of LaTeX, I haven't done any in a while. Would someone please tell me their thoughts? This was a major exam and I am pretty beat up..

 

[Zafa Pi]

You have 2 pencils and 3 pens. Each pencil and each pen is counted as an item. How many items do you have?

You have 2 pencils and 3 pens. Each pencil or each pen is counted as an item. How many items do you have?

I see these as the same question and same goes for your exam question.

 

[pbuk]

The question is badly worded.

The meaning of the sentence "Each flip and each roll is counted as a 'trial'" is "Each flip is counted as a 'trial' and each roll is counted as a 'trial'" (I assume 'trial' is in quotation marks in the original question to indicate that 'trial' is not used in its normal sense where trials are homogenous events with a single set of outcomes), so unless the question setter did not mean what was written then you have picked the wrong answer.

The sentence "Each flip or each roll is counted as a 'trial'" means "Each flip is counted as a 'trial' or each roll is counted as a 'trial'" which does not make sense, but if it had been worded "Each flip or roll is counted as a trial" it would have been clearer.

 

[mathman]

My reading of the original question. Flip coin until you get heads, then roll die until you get a six. The number of trials is the sum of the number of flips plus the number of rolls. What is the probability that this number is 5 (or less?).

 

[D H]

I concur with Zafa Pi and mathman. I don't see an ambiguity, and my interpretation is exactly what mathman said.

 

[mynameisfunk]

Ok guys, maybe you are right. I can see it now. If you were a professor, would you yield? I definitely see an ambiguity here. I just feel like it could have been much clearer as with these questions the wording needs to be pretty clear on what they want. In direct response to Zafa Pi, I definitely see now how those 2 statements are equivalent but one is definitely unambiguous and the other I feel like it COULD be interpreted as one pen and one pencil as a pair and are one item together.

 

[D H]

Students argue with their professors all the time. Who knows? You might be able to make a case here. Maybe.

 

[verty]

I remember in a final exam misinterpreting $8\sqrt{f(x)}$ as $\sqrt[8]{f(x)}$. I thought I did so well to answer that difficult question.

In your case, "flip" is singular, it can't refer to more than one flip. Each flip means, each one flip of the coin, it couldn't be n flips.