Bang or No Bang Game 8: Bang or No Bang

This game involves three balloons. Two of the balloons conain helium gas; one contains hydrogen. If a flame is held next to one of the balloons, then two things could happen. If it is helium inside the balloon, the balloon will burst, but there will be no "bang". Helium is what scientists call an inert gas; it doesn't explode in this situation. Hydrogen on the other hand is very reactive, and in the presence of a flame it will go BANG!

You have to guess which balloon you think will give the BANG. You have a one-in-three chance of being right, about 33%. You also have a two-in-three chance of being wrong.

The host (who knows where the BANG! is) then bursts one of the balloons you haven't chosen. one which they know is a "no bang". There are now only two balloons left; the one you chose and one other.

You are then offered the chance to change your mind, and swap to the other balloon. Should you?

The answer is: definitely yes! If you swap you double your chances of being right and of getting a BANG! "How can this be?" I hear you ask. If you don't believe me, play the game online here.

So why is this the case? Surely it is a 50:50 chance if there are two balloons left, so it shouldn't matter if you change your mind or not. This isn't true. Remember, when you first made your choice you had a one-in-three chance of being right, and a two-in-three chance of being wrong. This holds true no matter what happens after your initial choice. Once the host has popped a "no bang" balloon you still have your initial two-in-three chance of being wrong, so you should certainly swap.

If you're still not convinced, try to imagine the same game with one million balloons, only one of which is the BANG! You are asked to choose. The chance of you picking the right one is one-in-a-million. The host (who knows where the BANG! is) then bursts all but one balloon, and offers you the chance to swap.

What is most likely; that you picked correctly to start with (one-in-a-million) or that the host burst all but the BANG! balloon? You would be crazy not to swap in this instance.

It just goes to show that probability doesn't always seem right. When working out probability and risk, don't follow your instincts - work it out. Your instincts might be wrong!