This hand is adapted from  “Example 24” in from Hugh Kelsey’s book, Master the Odds in Bridge.  In hearts, North holds K3 , and South (declarer) holds A10985.  Declarer needs four tricks from the suit and can afford to lose the lead only once in his no-trump contract.  Therefore, he must capture one of the two missing honors.  If the suit breaks 3-3, the contract will be made, as one missing honor will drop and the other driven out. 

 Declarer plays the K and leads the 3. Declarer ponders whether he can succeed against a four-two split as an additional chance.  Success is inevitable if east held a doubleton honor, and failure is inevitable if west held four cards and both honors.  

 If South plays the 10, he can succeed if East holds both honors and four cards.  That is, one of the honors will drop on the third round.  If South rises with the A, the contract succeeds if West begins with a doubleton honor.   The point of this deal is that South does not need to analyze details about the positions for which failure or success is inevitable, but only needs to compare the odds for his two choices at the decision point. Neither choice affects the outcome of a 3-3  split.

South deduces that xx in West’s hand represents the 6 combinations from abcd : ab,ac,ad,bc,bd,cd. Alternatively, imagine drawing two cards from HHxxxx.  The chances of successively drawing two x’s are 4/6*3/5=6/15. 

Hx from West’s hand represents 4 possible x’s times two possible H’s, for  8 combinations of the 15 possible from HHxxxx.  The doubleton honor of HH was ruled out by the play of the king.  The probability of a given holding relates to the number of combinations available to form it.  Therefore, playing for the drop is favored by 8 to 6.