Have to also say that I was impressed that this was a well-formulated problem. I thought it must have an error, since the probability of burnout in the second 6 was greater than the probability of burnout in the first 6. But it actually works! Here's a neat way of verifying that - but if you aren't a GMAT instructor or a math buff, you can skip the following.
The probability that the bulb eventually burns out must equal 1, so we should have the following equation:
P(Burnout first 6) + P(Burnout second 6) + P(Burnout third 6) + ... = 1
The probability of not burning out during any given six months, without factoring in the probability that the bulb has survived up to that point, is easy to find: 2/3 for the first six months, 1/3 for the second six, 1/6 for the third six, 1/12 for the fourth six, etc. Similarly, the probabilities of burning out, again not factoring in the dependencies, are 1/3, 2/3, 5/6, 11/12, etc.
Now we bring in the history of the bulb. In order for it to burn out in the second six months, it must have survived the first six. So we have (Survive 1st)*(Not Survive 2nd), or the first number on the red list times the second number on the blue list. Likewise, for it not to burn out in the third six months, we need (Survive 1st) * (Survive 2nd) * (Not Survive 3rd), which is the product of the first TWO numbers on the red list and the third number on the green list.
This means our equation is
(Not Survive 1st) + (Survive 1st)(Not Survive 2nd) + (Survive 1st)(Survive 2nd)(Not Survive 3rd) + (Survive 1st)(Survive 2nd)(Survive 3rd)(Not Survive 4th) + ..., or
(1/3) + (2/3)(2/3) + (2/3)(1/3)(5/6) + (2/3)(1/3)(1/6)(11/12) + ..., or
(1/3) + (2/3)¹(1/2)�*(1 - (1/3)(1/2)�) + (2/3)²(1/2)¹(1 - (1/3)(1/2)¹) + (2/3)³(1/2)³(1 - (1/3)(1/2)²) + ..., or
Σ from k=0 to k=∞ of (2/3)� * (1/2)����¹��½� * (1 - (1/3)(1/2)��¹)
To my absolute astonishment, this sum converges to 1.
I've been teaching the GMAT for almost six years and answering student questions on random problems for nearly that entire time, and I don't think I've ever been this surprised. This problem looked so hinky that I would've bet the farm that it didn't work, but it does. (Good news for my farm!)
Whoever wrote this question, I owe you an apology and a cheap but tasty craft beer of your choice.
