Grinstead-Snell-Probability · Problem 4.3.3
Distributions and Densities
Mathematics · conditional-probability-paradoxes
Problem
In the preceding exercise, it is natural to ask ``How do we get the information that the given hand has an ace?'' Gridgeman considers two different ways that we might get this information. (Again, assume the deck consists of eight cards.) (a) Assume that the person holding the hand is asked to ``Name an ace in your hand'' and answers ``The ace of hearts.'' What is the probability that he has a second ace? (b) Suppose the person holding the hand is asked the more direct question ``Do you have the ace of hearts?'' and the answer is yes. What is the probability that he has a second ace?Intuitive explanation
The key insight is that how we learn about the ace matters enormously for conditional probability. In (a), the person actively chose which ace to name—if they had multiple aces, they could pick any one, so learning "ace of hearts" is less informative than it seems. In (b), we asked specifically about the ace of hearts, so a "yes" answer tells us exactly which card they hold. The different selection processes lead to different conditional probabilities, even though both scenarios involve having an ace of hearts.
Intuitive explanation
The key insight is that how we learn about the ace matters enormously for conditional probability. In (a), the person actively chose which ace to name—if they had multiple aces, they could pick any one, so learning "ace of hearts" is less informative than it seems. In (b), we asked specifically about the ace of hearts, so a "yes" answer tells us exactly which card they hold. The different selection processes lead to different conditional probabilities, even though both scenarios involve having an ace of hearts.