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Ever wonder why so many expert witnesses lead juries astray due to mathematical errors? Or why so many gamblers and investors are so bad at assessing relatively simple probability questions? First imagine that you consider yourself an expert (at something other than math), and then you encounter a question like this…

Imagine there’s a completely random event with two outcomes, say flipping a coin. Each flip has an equal probability of landing heads or tails. Now imagine that we’re interested in seeing how long it takes to get a certain sequence of outcomes.

Pattern 1

Tails, Heads, Tails

Pattern 2

Tails, Heads, Heads

Now, suppose we flip a coin until Pattern 1 is reached, note how many coin flips it took, and then we repeat the process many times and average how many flips it takes to get a tails-heads-tails sequence . After that, we go through the same process to see how many flips it takes to get Pattern 2, a tails-heads-heads sequence. For example if we start flipping a coin for pattern 1 and we see:

tails, heads, heads, tails, heads, tails

Then we reached Pattern 1 after only six coin tosses. Sometimes it will take as few as three coin tosses, but other times it will take many more. If we were to repeat this test thousands of times and calculate the average number of tosses it takes to get Pattern 1 and compare it to the average number of tosses it takes to get Pattern 2, which be the bigger number?

On average, which pattern takes fewer coin tosses?

  • They'll happen equally fast, on average. (78%, 1,775 Votes)
  • Tails, Heads, Tails takes fewer tosses! (11%, 258 Votes)
  • Tails, Heads, Heads takes fewer tosses! (9%, 213 Votes)
  • I can't figure it out. (2%, 36 Votes)

Total Voters: 2,282

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The first correct answer with a valid explanation wins a beer (if you can make it to Taipei to collect).
Update: Two correct answers are in! Ray Myers, with some lisp code to brute force the answer, and Robin with a clear explanation of why. When and if you make it out to collect, drinks at the Taiwan Beer Factory are on me.

Related Posts:
Even Simple Probabilty Puzzles Can Be Tricky
Game Theory and Bluffing
Good God are There a Lot of Morons on Digg

I’d been a bit worried that my Wednesday/Saturday class had deteriorated over the break for Spring Festival. It’s a pretty new class since it just opened in October. Also, quite a few kids in it were total beginners who didn’t even know the alphabet before starting. Since they can’t really practice spelling drills or phonics on their own, I was pretty nervous last night when I gave them a quiz. Fortunately it turned out alright. Here are the three parts to a quiz:

  1. Spelling- The students have to spell 10 words they’ve never heard before. I’ll read them as many times as the students want, though. Since this is a phonics drill rather than regurgitating words they’ve previously memorized, I obviously have to accept phonetic equivalents. For example, if I say “note” and a student writes “noat”, it’s correct. The same goes for “caper” and “kaper”, etc… Sometimes, when I want students to use a certain spelling rule, I’ll tell them NOT to use a certain letter. For example, if I say “sell”, they’ll probably write “sell”. But if I say, “Don’t use s,” then the students have to write “cell”. This section is very hard for most students, especially students who have previously spent years learning from non-native teachers with poor pronunciation. Spelling half of the words correctly counts as a pass. Students who fail this section have extra spelling drill homework until the next quiz.
  2. Sentence Answers- I ask two questions. The students write down the answers. They get 罰寫 for mis-spelled words and for grammar mistakes.
  3. Comprehension Phrases- One important part of our curriculum is comprehension phrases that the students have to understand, but don’t have to be able to say. These are things that I would otherwise have to say in Chinese all the time, mostly classroom commands. These phrases aren’t really important enough that it’s worth the class time to get the students to learn how to say them at the lower levels. It is still vitally important that the students understand these phrases, though. Getting less than 2/3 on this section results in 罰錄.

Here are two quizzes. The one on the left is a D student’s quiz, and the one on the right is an A student’s quiz.

Here are the same two quizzes graded:

As you can see, the D student made a few mistakes related to spelling rules and others were listenning problems. She spelled the first syllable of “hickory” as “heek”. This is one of the most common problems I see. Many, many students who study at big chains before coming to my school have problems differentiating between short i’s and long e’s. Word pairs such as “sit”/”seat” and “his”/”he’s” trouble them for a very, very long time. Long a’s and short e’s are probably the 2nd most common problem. Examples of this problem would be “special”/”spatial”, “seven”/”saivun”, “tech”/”take”, and so on. Why is it that my students with MORE English exposure have worse listening abilities than those who came with nothing? I can’t say for sure, but I’m venturing a guess. It’s because their previous teachers had some pretty serious phonics problems of their own. In any case, it’s not the students’ fault and eventually, I will get their phonics fixed.