Shuffle the three cards and draw one without looking.
Your tally
- Rounds
- 0
- Obama facing up
- 0
- Obama on both sides
- 0
Draw
00
No draws yet. Choose how many cards to draw.
Estimate after each Obama-up draw
Every draw so far
- Obama facing up, Obama on the other side 0
- Obama facing up, Gaga on the other side 0
- Gaga facing up, so the draw doesn’t count 0
Reveal the answer
The tempting answer is 1/2: the card is either Obama/Obama or Obama/Gaga, so it seems to be a coin flip between them. But seeing Obama is not equally likely for those two cards. Count sides instead of cards.
Gaga/Gaga
Obama/Gaga
other side Gaga
other side Gaga
Obama/Obama
other side Obama
other side Obama
other side Obama
other side Obama
Each of the six sides is equally likely to be the one facing up. Three of them show Obama, and two of those three belong to the Obama/Obama card. So the probability that Obama is on the other side is 2/3.
In symbols, let F be the event that Obama is facing up and B the event that he is on the other side. Then
P(B | F) = P(B ∩ F)P(F) = 1/31/2 = 2/3.