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Three Cards (illustration)
FILE No.140Unsolved

Three Cards

Difficulty ★★★about 5 minLogic PuzzleProbabilityCounting cases
Cards: red on both sides, white on both sides, and red/white. One was drawn and the top face is red. What is the probability the back is red too?

Case Record

There are three cards: red on both sides, white on both sides, and red on one side and white on the other. You draw one with your eyes closed and place it on the table, in a random orientation. The top face is red.

Your question

What is the probability that the back of that card is also red?

Case Files

Diagram

Three cardsRedRedCard 1WhiteWhiteCard 2RedWhiteCard 3One card is drawn and put down at random: the top face is red.What is the chance the other side is red too?
Diagram

Hints

If you get stuck, open the hints one at a time, in order.

Open hint 1

How many red "faces" are there in all?

Open hint 2

Count the two red faces of the same card separately.

Open hint 3

What are the 3 ways for the top to be red?

Solve It

Your question

What is the probability that the back of that card is also red?

Choose the answer closest to the truth.

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The Truth

2/3

The answer is "2/3."

There are 3 red faces in all (two from the double-red card, one from the red/white card). Of the 3 ways the top can be red, the back is also red in the 2 ways that belong to the double-red card.

1/2 comes from thinking "it's one of the remaining two cards," which overlooks how the red faces are counted.

The three ways the top face can be redRed face on topBackBack red?Card 1, red side ARed (Card 1, side B)YesCard 1, red side BRed (Card 1, side A)YesCard 3, red sideWhiteNo2 of the 3 ways. Answer: 2/3
Diagram of the solution

Key Points

  • Count "faces," not cards
  • Split into equally likely cases
  • Doubt your intuition of 1/2

Why people get fooled

Because you think "one of the remaining two cards" and say 1/2.

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