# 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 cheapest discard you can make on the Maestro’s ♣A is ♣J. This will bring the Maestro’s trick point total to 27+11+2 = 40. 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:(40 points)

♠ Q

♥ Q

♣ Q

♦ AK

You:(18 points)

♠ A

♥ AJ

♣ TK

♦ —

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 ♥J,
cash ♥A, and then cash your remaining 3 cards in any order.
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 more than 33 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 18+11+11 = 40. 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 five cards you haven’t yet seen: ♠Q, ♥Q, ♣Q, or ♦AK. We can divide them into 3 cases:

**Case 1:** If you draw any of those three red cards, then you know
the Maestro holds both ♠Q and ♣Q. In that case, you can
cash both ♠A and ♣T, which will bring your trick point
total to 40+11+10+3+3 = 67. This is enough, and you will win 2 game
points, since the Maestro only has 27 trick points.

**Case 2:** If you draw ♣Q, you will be on lead from this position:

Maestro:(27 points)

♠ Q

♥ QJ

♣ —

♦ AK

You:(40 points)

♠ A

♥ —

♣ TKQJ

♦ —

Even though the Maestro holds all the remaining trumps, the 20-point marriage you just drew will be enough for you to win. Lead ♠A, capturing the Maestro’s ♠Q, and then declare the marriage. This will bring your trick point total to 40+11+3+20 = 74, and you will again win 2 game points.

**Case 3:** If you draw ♠Q, you will be on lead from this position:

Maestro:(27 points)

♠ —

♥ QJ

♣ Q

♦ AK

You:(40 points)

♠ AQ

♥ —

♣ TKJ

♦ —

♠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 ♣T, capturing the Maestro’s ♣Q, but that only brings your trick point total to 40+10+3 = 53. 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 ♥Q and immediately lead his ♣Q to force out your ♣T. This will leave you on lead from this position:

Maestro:(33 points)

♠ —

♥ J

♣ —

♦ AK

You:(53 points)

♠ A

♥ —

♣ KJ

♦ —

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 three cases above is that there are 4 cards
you can draw from the stock (♥Q, ♦AK, ♣Q) that will result in a
gain of 2 game points for you, and 1 card you can draw (♠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) = ⅘, p(−1) = ⅕. This means the
expected value of *X* is E[*X*] = ⅘(+2) + ⅕(−1) = 7/5 = **1.4**.

*(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 1.4 game points. (You will probably win 2 game points, but 1/5 of the time you will lose 1 game point.) You should therefore trump his ♦A, for the greater expected gain.

“Very, very nicely played!” the Maestro exclaims when the deal is over and he records 2 game points for you. “I can see that you understand how analyzing the expected game points leads you to the best line of play.”

There are lots of other Schnapsen Log columns posted since May 2012 that use this sort of expected game point analysis.

© 2015 Martin Tompa. All rights reserved.