You are probably familiar with Randall Munroe of XKCD fame. One of his most famous strips is this:
(image © Randall Munroe)
I am usually good about not falling into this trap, but sometimes the bait is too tempting, and I blow an hour or two futilely trying to educate someone – usually with a smile on my virtual face, since I’m a honey-not-vinegar kind of guy.
Yesterday I stuck my foot directly into the bear trap, and (unsurprisingly) it snapped shut. In my defense there was a mathematical riddle just sitting there on the tension plate. What, I was supposed to leave it there?
The riddle was posted innocently to a D&D Facebook group. It’s an old riddle, long-since solved, but the solution is not intuitive, and there are plenty of smart people out there who insist upon a wrong solution. Here it is:
“You have three bags, each containing two gems. The first bag contains two blue gems, the second bag contains two red gems, and the third bag contains one blue gem and one red gem.
You pick a random bag and take out one gem.
It is a blue gem.
With what certainty would you guess that the remaining gem from the same bag is also blue?”
That’s it verbatim as it appeared in the Facebook thread. I think traditionally the riddle uses marbles, but this is a D&D group, so we’re probably looking at citrines or tourmalines or something.
The obvious (but wrong) answer is 50% likely.
The less obvious (but correct) answer is 66% likely.
The way to think about it actually quite simple:
- You have three bags with two gems each, which means when you pick a single gem, there are six possible ways that you can do it:
a. Draw one of the red gems from the bag with two red gems
b. Draw the other red gem from the bag with two red gems
c. Draw the blue gem from the bag that has one of each color
d. Draw the red gem from the bag that has one of each color
e. Draw one of the blue gems from the bag with two blue gems
f. Draw the other blue gem from the bag that has two blue gems
- In the riddle, we know you’ve picked out a blue gem, which means that a, b, and d didn’t happen.
- That means there are possible choices you DID make, each with the same likelihood: c, e, and f.
- If your draw was e or f, then the remaining gem from your bag is blue. If your draw was c, the remaining gem is red. Therefore, Q.E.D., there is a 66% likelihood that, given your first gem was blue, that the other gem in the bag is also blue.
I explained this in about four different ways. I even re-taught myself Python so I could write a simple script to simulate the problem. Sure enough: 66%. And yet there’s one fellow on this thread who has made a personal crusade out of his certainty that the answer is 50%. (I’ll leave him anonymous; I harbor no ill will towards him.) He has gone so far as to post YouTube videos showing how the puzzle can be “solved” with (I kid you not) collapsing quantum states of the gem bags. And, no, he’s not trolling the discussion. I’ve been around the block enough times to recognize a bridge under-dweller, and this guy is legit.
He has not convinced me, of course, that 50% is the correct answer. Because it’s not. What he has convinced me of is that no power on this world or any other will convince him he’s wrong.
The funny thing is, I’m sure he’s thinking the exact same thing about me.