September 2021 - 28

Math in Bridge
Suppose your partner opens a 15-17 1NT and you have
8 HCP, no five-card or longer major, and no six-card or
longer minor. You have a tough decision to make. Many
players will decide whether they consider it a " good " 8
or a " bad " 8 by looking at intermediates (10s and 9s), suit
length (a five-card minor might be a source of tricks),
honor quality (bridge wisdom says that aces are really
worth more than 4 points and jacks worth less than 1) and
placement (Q-J in the same suit is worth more than Q
and J in separate suits). With a good 8 they might invite
3NT or check for a major fit, whereas with a bad 8 they
might pass. One danger in inviting is that you might end
up in 2NT when the limit of the hand is seven tricks.
I'd like to bring up another point to bear in mind. Presumably
your partner will accept your invitation with 17
points and decline with 15. With 16 points, he, too, might
look at how good those points are. Is he equally likely to
hold 15, 16, or 17 points? The answer is no. According to
a table found online, the probability of holding 15 HCP
is 4.42%, the probability of holding 16 HCP is 3.31%, and
the probability of holding 17 HCP is 2.36%. Combining
these, we see that the probability of holding 15-17 points
is 10.09%. Dividing each of the original probabilities by
this figure, we find that the relative chances of 15, 16, or
17 HCP are about 44%, 33%, and 23%, respectively. So almost
half the time when partner opens 1NT, he has just 15
points! (These calculations are off a little, because we are
not using the information that partner's hand is balanced.
I doubt that this significantly distorts the odds, however.)
We can make similar kinds of calculations in other situations.
Suppose you hold:
Q 4 kK 6 2 lQ 9 6 5 '9 8 7 2.
Partner opens 1, you bid a forcing 1NT, and partner
rebids 2k. Ignoring the possibilities that he has six or
more spades, or longer hearts than spades (with five
spades and six hearts and a hand not strong enough to reverse,
he would have to open 1), there are two possibilities
for the length of his heart suit: four and five. Should
you take a false preference to spades or pass?
Bridge wisdom is that a 5-2 fit plays better than a 4-3
fit. But if partner is 5-5, you'd certainly prefer to be in
28 | Bridge Bulletin | September 2021
BY JERROLD GROSSMAN
Using probability to assess partner's holding
hearts. Our instincts tell us that four is more likely than
five. Can we compute how much more likely?
Here's my analysis. Using tables of hand distribution
probabilities available online, or computing them from
scratch mathematically, we get the following probabilities:
5=4=2=2
5=4=3=1
or 5=4=1=3
5=4=4=0 or 5=4=0=4
5=5=2=1 or 5=5=1=2
5=5=3=0 or 5=5=0=3
0.88%
1.08%
0.21%
0.53%
0.15%
This comes to a total of about 2.17% for partner's holding
four hearts, versus 0.68% for five, meaning that he is
more than a 3-to-1 favorite to hold only four hearts.
With a likelihood of landing in a seven-card fit in either
major, you should give a preference for spades. If we
throw in the case of six spades and four hearts, assuming
partner would always choose to rebid hearts instead of
repeating spades, which occurs with probability 0.50%, it
tilts the odds in favor of bidding 2 even more.
Here's one more miscellaneous calculation that might
come in handy. You hold 'A K Q 5 4 3 2 with no outside
entries and are considering opening a gambling 3NT in
first seat. How likely is it that you will score seven club
tricks at notrump? You will fail to bring home the suit if
one opponent holds four or more clubs, unless partner
has 'J x. Using the appropriate mathematical formulas,
I compute that this will happen about 15% of the time; in
other words, you'll be successful about 85% of the time.
If your holding is 'A K Q 10 4 3 2, then your chances of
success rise to about 89%, and with 'A K Q J 4 3 2 you're
a 98% favorite.
If you have mathematical issues you'd like to see addressed
in this column, let me know.
Jerrold Grossman is a retired professor of mathematics at Oakland
University in Michigan and an Emerald Life Master. He writes about
how to use mathematics in making bridge decisions. Bridge math
questions or topic suggestions can be sent to
grossman@oakland.edu.
September 2021

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