# 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 ♣Q, so that he does not win any more club tricks. This trick will bring the Maestro’s trick point total to 28+11+3 = 42. The Maestro will draw the last face-down card from the stock and you will draw the face-up ♦J, which puts the Maestro on lead from this position:

Maestro:(42 points)

♠ KJ

♥ T

♣ T

♦ Q

You:(20 points)

♠ T

♥ AJ

♣ —

♦ KJ

From here, whatever the Maestro does, you can take the rest of the
tricks. All you have to do is use ♦K to pull the Maestro’s
last trump and, after that, all your cards are winners. For example,
if he leads ♣T from the diagrammed position, you trump with ♦J,
cash your three winners ♦K, ♥A, and ♠T, 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 42 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 ♦K. This will bring your trick point total to 20+11+4 = 35. Since you won the trick, you will draw the random, face-down card from the stock and the Maestro will draw the face-up ♦J. You could draw any of the four cards you haven’t yet seen: ♠K, ♠J, ♥T, or ♣T. We can divide them into 2 cases:

**Case 1:** If you draw any of those three black cards, then you know
the Maestro holds ♥T and at least one spade. In that case, you can
cash ♥A and ♠T, which will bring your trick point total to
at least 35+11+10+10+2 = 68. This is enough, and you will win 2 game
points, since the Maestro only has 28 trick points.

**Case 2:** If you draw ♥T, you will be on lead from this position:

Maestro:(28 points)

♠ KJ

♥ —

♣ T

♦ QJ

You:(35 points)

♠ T

♥ ATJ

♣ Q

♦ —

♥T 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 ♠T, capturing the Maestro’s ♠J, but that only brings your trick point total to 35+10+2 = 47. 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 ♥J from the diagrammed position. The Maestro will trump with ♦Q and immediately lead his ♠J to force out your ♠T. This will leave you on lead from this position:

Maestro:(33 points)

♠ K

♥ —

♣ T

♦ J

You:(47 points)

♠ —

♥ AT

♣ Q

♦ —

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 (♠K, ♠J, ♣T) that will result in a gain
of 2 game points for you, and 1 card you can draw (♥T) 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) = ¾, p(−1) = ¼. This means
the expected value of *X* is E[*X*] = ¾(+2) + ¼(−1) = 5/4 =
**1.25**.

*(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 1.25 game points. (You will win 2 game points 3/4 of the time, but you will lose 1 game point 1/4 of the time.) You should therefore trump his ♣A, for the greater expected gain.

“Very nicely played!” the Maestro exclaims when the deal is over and he records 2 game points for you. “Trumping my ♣A was a good gamble. I 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.

© 2017 Martin Tompa. All rights reserved.