Game of Hearts: Addendum

If, when you are to pass cards, you hold four spades to the Ace or the King (or both the Ace and the King), should you pass the high spades to avoid having to take the Queen or should you improve your hand by passing cards in other suits?

Modified probability considerations suggest that, when holding four spades with the Ace or King, you should hold on to all the spades and pass other cards. The probability is over 90% that you will not be forced to take the Queen but this result is more due to bad play by opponents than it is to objective probabilities. Notice the behavior of opponents when the Queen is involved, as described above.

Here are very rough estimates of the conditions that lead to success in avoiding the capture of the Queen:

• Remaining spades are equally divided among opponents. Each opponent has three spades and therefore they can lead spades only three times and cannot force your Ace or King to fall. {Rough estimate of probability: 1/84 or .012.)
  
  
• Player to left leads a spade early enough that Ace can be safely played. (Rough estimate: over 1/3 or .333.)
  
  
• Player opposite leads a spade, finessing Queen held by player on your right. You then luckily and safely play the Ace or King. (Roughly 1/9 or .111.)
  
  
• You are, or become, void in a suit that, when led, permits discarding the Ace. (Rough estimate: 1/20 or .05.)
  
  
• You receive the Queen and can get rid of it. (Probability unknown.)
  
  
• You are dealt the Queen when it is time for a passing of cards to the right. You pass the Queen. Thus, on any lead of spades, excepting the lead of the Queen, you can safely play the Ace or King. (Rough probability: 1/16 or .0625.)

The sum of the estimated probabilities is .718, which appears to be too high. However, in an actual experiment of 17 games consisting of 179 hands, success in avoiding the Queen was achieved 19 out of 20 times or, .95 or 95% of the time. Again, there were in the 179 hands only 20 where the player held four spades to the Ace or King. Bad play by opponents is an important cause for the discrepancy between estimated probabilities and experimental results.

Copyright © Clement Washburn 2003