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
Famous quotes containing the words card and/or version:
“Mothers are not the nameless, faceless stereotypes who appear once a year on a greeting card with their virtues set to prose, but women who have been dealt a hand for life and play each card one at a time the best way they know how. No mother is all good or all bad, all laughing or all serious, all loving or all angry. Ambivalence rushes through their veins.”
—Erma Bombeck (20th century)
“Remember that you were a slave in the land of Egypt, and the LORD your God brought you out from there with a mighty hand and an outstretched arm; therefore the LORD your God commanded you to keep the sabbath day.”
—Bible: Hebrew, Deuteronomy 5:15.
See Exodus 22:8 for a different version of this fourth commandment.