### The Effect of Added Information on Probabilities

Suppose I deal two cards face down from an eight-card deck containing the four Aces and the four Kings. What is the probability that both cards are Aces? There are 28 possible hands and 6 of them have two Aces, so the answer is 6/28 = 0.21. This is illustrated in the 1st column of the table below.

Now suppose I deal two cards, look at them, and tell you that at least one is an Ace. What’s the probability I have 2 Aces? The 6 hands that just have Kings are no longer possible, so the answer is 6/22 = 0.27. This is shown in the 2nd column of the table.

This time, after I look at the cards, I tell you that I have a black Ace. Now the answer is 5/13 = 0.38. See column 3.

Finally, after I look at the cards, I tell you that I have the Ace of Spades. Now the answer is 3/7 = 0.43. See column 4.

To see an interesting similar problem, Google ‘girl named florida problem’.

The 28 possible hands are shown in the table. Green cells contain 2 Aces. Gray cells are not possible given the available information.

The image below shows the calculations for a 5 card hand dealt from a full deck. (Epstein, Richard, The Theory of Gambling and Statistical Logic) No Info
1 Ace
Black Ace
A♠
A♠ A♥
A♠ A♥
A♠ A♥
A♠ A♥
A♠ A♦
A♠ A♦
A♠ A♦
A♠ A♦
A♠ A♣
A♠ A♣
A♠ A♣
A♠ A♣
A♠ K♠
A♠ K♠
A♠ K♠
A♠ K♠
A♠ K♥
A♠ K♥
A♠ K♥
A♠ K♥
A♠ K♦
A♠ K♦
A♠ K♦
A♠ K♦
A♠ K♣
A♠ K♣
A♠ K♣
A♠ K♣
A♥A♦
A♥A♦
A♥A♦
A♥A♦
A♥A♣
A♥A♣
A♥A♣
A♥A♣
A♥K♠
A♥K♠
A♥K♠
A♥K♠
A♥K♥
A♥K♥
A♥K♥
A♥K♥
A♥K♦
A♥K♦
A♥K♦
A♥K♦
A♥K♣
A♥K♣
A♥K♣
A♥K♣
A♦A♣
A♦A♣
A♦A♣
A♦A♣
A♦K♠
A♦K♠
A♦K♠
A♦K♠
A♦K♥
A♦K♥
A♦K♥
A♦K♥
A♦K♦
A♦K♦
A♦K♦
A♦K♦
A♦K♣
A♦K♣
A♦K♣
A♦K♣
A♣K♠
A♣K♠
A♣K♠
A♣K♠
A♣K♥
A♣K♥
A♣K♥
A♣K♥
A♣K♦
A♣K♦
A♣K♦
A♣K♦
A♣K♣
A♣K♣
A♣K♣
A♣K♣
K♠K♥
K♠K♥
K♠K♥
K♠K♥
K♠K♦
K♠K♦
K♠K♦
K♠K♦
K♠K♣
K♠K♣
K♠K♣
K♠K♣
K♥K♦
K♥K♦
K♥K♦
K♥K♦
K♥K♣
K♥K♣
K♥K♣
K♥K♣
K♦K♣
K♦K♣
K♦K♣
K♦K♣