# The Schnapsen Log

## Homework on Expected Values (solution)

#### Martin Tompa

*(a) How will the deal play out if you duck this trick?
Who will win, and how many game points?*

The best discard you can make on the Maestro’s ♦A would be ♦J, so that he does not win any more diamond tricks. This trick will bring the Maestro’s trick point total to 21+11+2 = 34. The Maestro will draw the last face-down card from the stock and you will draw the face-up ♥Q, which puts the Maestro on lead from this position:

Maestro:(34 points)

♠ K

♥ T

♣ Q

♦ TK

You:(17 points)

♠ AJ

♥ AQ

♣ A

♦ —

From here, whatever the Maestro does, you can take the rest of the
tricks. All you have to do is use ♥A to pull the Maestro’s
last trump and, after that, all your cards are winners. For example,
if he leads ♦K from the diagrammed position, you trump with ♥Q,
cash your three aces, and then your remaining ♠J is also a
winner. Since you will take the last trick, it doesn’t even matter
whether or not you reach 66 trick points. **You will win 1 game
point**, since the Maestro already has 34 trick points himself.

*(b) How will the deal play out if you win this trick?*

The only way you can win the trick in which the Maestro led ♦A is by trumping with ♥A. This will bring your trick point total to 17+11+11 = 39. Since you won the trick, you will draw the random, face-down card from the stock and the Maestro will draw the face-up ♥Q. You could draw any of the five cards you haven’t yet seen: ♠K, ♥T, ♣Q, ♦T, or ♦K. We can divide them into 2 cases:

**Case 1:** If you draw any of those three red cards, then you know
the Maestro holds both missing black cards, ♠K and ♣Q. In that
case, you can cash your two black aces, which will bring
your trick point total to 39+11+11+4+3 = 68. This is enough, and you
will win 2 game points, since the Maestro only has 21 trick points.

**Case 2:** If you draw either of the unseen black cards, ♠K or ♣Q,
you will be on lead from this position (which assumes you drew ♣Q)
or one very similar (if you draw ♠K):

Maestro:(21 points)

♠ K

♥ TQ

♣ —

♦ TK

You:(39 points)

♠ AJ

♥ —

♣ AQ

♦ J

Either of those small black cards, ♠K or ♣Q, is an unlucky draw for you, and there is nothing you can do to win from this position. The only trick you can possibly win is ♠A, capturing the Maestro’s ♠K, but that only brings your trick point total to 39+11+4 = 54. You also have no chance to win the last trick against someone of the Maestro’s skill, because of his two trumps. For instance, you might try leading ♣Q from the diagrammed position. The Maestro will trump with ♥T and immediately lead his ♠K to force out your ♠A. This will leave you on lead from this position:

Maestro:(34 points)

♠ —

♥ Q

♣ —

♦ TK

You:(54 points)

♠ J

♥ —

♣ A

♦ J

All the remaining trick are the Maestro’s, and he will win 1 game point.

*Combine these appropriately to determine the expected number of game
points that you will win.*

What we have seen from the two cases above is that there are 3 cards
you can draw from the stock (♥T, ♦T, ♦K) that will result in a gain
of 2 game points for you, and 2 cards you can draw (♠K, ♣Q) that
will result in a loss of 1 game point for you. If we let the random
variable *X* be the number of game points you win, its probability
mass function is therefore p(+2) = ⅗, p(−1) = ⅖. This means
the expected value of *X* is E[*X*] = ⅗(+2) + ⅖(−1) = 4/5 =
**0.8**.

*(c) Based on your answers, will you duck the Maestro’s ♦A or trump
it? Why?*

If you duck, the answer to part (a) says that you are guaranteed to win 1 game point. If you instead trump, the answer to part (b) says that you expect to win 0.8 game points. (You will win 2 game points 3/5 of the time, but you will lose 1 game point 2/5 of the time.) You should therefore duck his ♦A, for the greater expected gain.

“Very nicely played!” the Maestro exclaims when the deal is over and he records 1 game point for you. “Ducking my ♦A was a nice safety play, avoiding drawing one of those black cards from the stock. I can see that you understand how analyzing the expected game points leads you to the best line of play.”

For another Schnapsen problem with a similar analysis, see the column It All Depends. There are lots of other Schnapsen Log columns posted since May 2012 that use this sort of expected game point analysis.

© 2016 Martin Tompa. All rights reserved.