The Puzzle of HHT versus HTH

Are you open minded?  A classroom experiment I witnessed some years ago suggests probably not.  In this blog I am discussing subjects that may not have widespread agreement, such as 3 speaker versus 2 speaker playback, or thoughts about analog and digital audio.  And I wonder whether people really read and think about what I write, or if they ignore new information or ideas that might disagree with their existing beliefs.

More than 20 years ago, I attended a class by Professor Ron Howard at Stanford on “Decision Analysis” – I just checked and he still has a web page at Stanford so I guess he’s still teaching: https://engineering.stanford.edu/profile/rhoward.

In one lecture, Professor Howard posed a problem which, if I recall correctly, was titled something like “HHT versus HTH.”  The problem Howard posed is interesting because, in the classroom setting, and without a few quiet moments to reflect and think it through, it is a difficult one for most students to solve, including the bright master’s degree level students at Stanford that were the bulk of the class.  Therefore, and this is what was so fascinating, what is actually a question with a definite right/wrong answer takes on the characteristics of a subjective issue where people can “agree to disagree.”

Here’s the problem: two students (A and B) are brought to the front of the class, and each one is given a medallion with 2 sides – one side is “H” (head) and the other “T” (tails), and assume that in tossing the medallion the probability of H or T is equal.

Student A is to keep tossing the medallion until the sequence H then H then
T is achieved. 

Student B is to keep tossing the medallion until the sequence H then T then H is achieved. 

Question: who do you expect to achieve their goal in the least number of tosses: Student A or Student B or is it equal?

Of course, the interesting thing here is NOT the question, but the classroom
activity.  What Prof. Howard did after posing the question was to poll the class – how
many think A? how many think B? how many equal? how many are undecided? The
votes were recorded.  Then he asked a student in the audience to explain why “A” and another student to explain why “B”, etc.  Then he polled the class again, and again recorded the votes.The above process of listening to student explanations, polling the class, and recording the votes was repeated several times.  During the process, a lot of absurd and wrong statements were made, but eventually, one student figured out the problem, and stated the correct answer with a clear, logical explanation.

You might think that after the student stated and explained the correct answer, all the votes would switch.  But that did not happen.  The interesting thing?  Nobody listened.  That’s right, the votes remained fairly stable – people made up their minds initially, and then paid basically no attention to anything that was said.

Listening to the lecture and classroom participation/response was a
fascinating experience for me. Oh, and for those who care, it’s student B
who would be expected to first achieve the goal sequence.  Quick explanation: the two sequences have different symmetry properties, so that if we suppose both A and B miss on the second toss, A (who tossed HT) needs the third toss to begin his sequence, but B (who tossed HH) can start his sequence with the second toss, so B needs fewer tosses on average to achieve his sequence.

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