# 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 and most reasonable discard you can make on the Maestro’s ♦A would be ♣J. This trick will bring the Maestro’s trick point total to 20+11+2 = 33. 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:(33 points)

♠ K

♥ KJ

♣ AT

♦ —

You:(19 points)

♠ AJ

♥ AT

♣ —

♦ Q

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 ♣T 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 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 19+11+11 = 41. 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: ♠K, ♥K, ♥J, ♣A, 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 both missing hearts, ♥KJ. In that case, you can
run ♥AT, which will bring your trick point total to
41+11+10+4+2 = 68. This is enough, and you will win 2 game points,
since the Maestro only has 20 trick points.

**Case 2:** If you draw either of the unseen hearts, ♥KJ, you will be on
lead from this position (which assumes you drew ♥J) or one nearly
identical (if you draw ♥K):

Maestro:(20 points)

♠ KJ

♥ K

♣ AT

♦ —

You:(41 points)

♠ —

♥ ATJ

♣ J

♦ Q

Either of those small hearts ♥KJ 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 41+11+4 = 56. 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 ♠K and immediately lead his ♥K to force out your ♥A. This will leave you on lead from this position:

Maestro:(27 points)

♠ J

♥ —

♣ AT

♦ —

You:(56 points)

♠ —

♥ TJ

♣ 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 (♠K, ♣A, ♣T) that will result in a gain
of 2 game points for you, and 2 cards you can draw (♥K, ♥J) 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 low hearts 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.

© 2015 Martin Tompa. All rights reserved.