Card Version
Suppose there are three cards:
- A black card that is black on both sides,
- A white card that is white on both sides, and
- A mixed card that is black on one side and white on the other.
All the cards are placed into a hat and one is pulled at random and placed on a table. The side facing up is black. What are the odds that the other side is also black?
The answer is that the other side is black with probability 2⁄3. However, common intuition suggests a probability of 1⁄2 either because there are two cards with black on them that this card could be, or because there are 3 white and 3 black sides and many people forget to eliminate the possibility of the "white card" in this situation (i.e. the card they flipped CANNOT be the "white card" because a black side was turned over).
In a survey of 53 Psychology freshmen taking an introductory probability course, 35 incorrectly responded 1⁄2; only 3 students correctly responded 2⁄3.
Another presentation of the problem is to say : pick a random card out of the three, what are the odds that it has the same color on the other side ? Since only one card is mixed and two have the same color on their sides, it is easier to understand that the probability is 2⁄3. Also note that saying that the color is black (or the coin is gold) instead of white doesn't matter since it is symmetric: the answer is the same for white. So is the answer for the generic question 'same color on both sides'.
Read more about this topic: Bertrand's Box Paradox
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