EXAMPLE 4.42

Probability of being dealt a heart. Doyle is a professional poker player. He stares at the dealer, who prepares to deal. What is the probability that the card dealt to Doyle is a heart? There are 52 cards in the deck. Because the deck was carefully shuffled, the next card dealt is equally likely to be any of the cards that Doyle has not seen. Thirteen of the 52 cards are hearts. So

This calculation assumes that Doyle knows nothing about any cards already dealt. Suppose now that he is looking at four cards already in his hand and that they are all hearts. He knows nothing about the other 48 cards except that exactly nine (13 − 4) hearts are among them. Doyle’s probability of being dealt a heart given what he knows is now

Knowing that there are four hearts among the four cards Doyle can see changes the probability that the next card dealt is a heart.