To remind you, reCAPTCHA asks you to decipher two smeared words before you can register for, say, a gmail account.  One of the words is being used to test whether you are a human and not a computer.  The reCAPTCHA system knows the right answer for that word and checks whether you get it right.  The reCAPTCHA system doesn’t know the other word and is hoping you will help figure it out.  If you get the test word right, then your answer on the unkown word is assumed to be correct and used in a massive parallel process of digitizing books.  The words are randomly ordered so you cannot know which is the test word.

Once you know this, you many wonder whether you can save yourself time by just filling in the first word and hoping that one is the test word.  You will be right with 50% probability.  And if so, you will cut your time in half.  If you are unlucky, you try again, and you keep on guessing one word until you get lucky.  What is the expected time from using this strategy?

Let’s assume it takes 1 second to type in one word.  If you answer both words you are sure to get through at the cost of 2 seconds of your time.  If you answer one word each time then with probability 1/2 you will pass in 1 second, with probability 1/4 you will pass in 2 seconds, probability 1/8 you pass in 3 seconds, etc.    Then your expected time to pass is

\sum_{t=1}^\infty \frac{t}{2^t}

Is this more or less than 2?  Answer after the jump.

D’oh!

\sum_{t=1}^\infty \frac{t}{2^t} =2

and if you add in the small delay before the next pair words appears, you are better off taking the sure thing (unless you have declining marginal opportunity cost of time.)