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.

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