Solution: Coloured Cards Puzzle

The Puzzle:

Three cards are placed in a hat. They are

• A card that is coloured red on both sides.
• A card that is coloured green on both sides.
• A card that is coloured green on one side and red on the other side.

One card is drawn at random from the hat and placed with the green side up on the table (when one side of a card is face up, you can't see what's on the other side). What is the probability that the other side of the card is also green?

The Solution:

It's very tempting to say that the probability is \(\dfrac{1}{2}\), but this is incorrect. This is actually a conditional probability problem: we're looking for the probability that the card that has been drawn is the card that is green on both faces, given that the first face is green. Using the formula for conditional probability, this is \(\dfrac{1 \times \dfrac{1}{3}}{\dfrac{1}{2}} = \dfrac{2}{3}\).

I have to admit, this still makes my brain hurt. Let's try an intuitive explanation. Once the card is drawn from the hat and placed face upwards, we know that it is not the card that is coloured red on both sides. So, there are three possibilities for the second face of the card that has been drawn:

• It is the bottom face of the card that is green on both sides.
• It is the top face of the card that is green on both sides.
• It is the red face of the card that is red on one side and green on the other.

In two of these three cases, the other side of the card will be green, so the probability that the second face is green is \(\dfrac{2}{3}\).