Excerpt from Probability in Ancient Times; or, Shall I Go Down after the Philistines?
by Janet S. Milton
from the
Mathematics Teacher
March 1989
This article explores an interesting application of
combinations, geometric series, and elementary
probability that arises in a Biblical setting. I
have used this example in courses in finite
mathematics, mathematical statistics, and elementary
calculus. Its nontraditional nature seldom
fails to arouse the curiosity of students.
In Old Testament times, the three main
channels of God's revelation were the
sages, the priests, and the prophets. The
priests had various duties, but their primary
function until after the death of King
Solomon was that of divining the will of
God by the use of "Urim and Thummim."
Scholars think that these words refer to objects,
perhaps stones, used by the high
priest to ascertain the will of God in any
important matter affecting the nation. One
theory is that the stones were used to cast
lots, with the manner of their falling somehow
revealing the Lord's will.
Biblical scholar Horace R. Weaver
gives this general description of the
manner in which Urim and Thummim were
used. The sacred container used for divining
Urim and Thummim was a small box.
"Yes-no" type questions were brought to
the priest. He would place six stones, three
white and three black, in the sacred container.
The black stones might represent
"no" and the white, "yes." The stones were
mixed and three drawn at random. If all
three were white then God's answer to the
question was "yes"; three blacks implied a
"no" answer; and a mix indicated that God
gave no response that day.
For example, in 1 Samuel 14:37 Saul
asks, " Shall I go down after the Philistines?
Wilt thou deliver them into the
hand of Israel?" The King James version of
the Old Testament states, "He answered
him not that day" : the stones, when drawn,
were mixed in color. Other references to
this sort of decision making are found in
Exodus 28:90, where it states "and thou
shalt put in the breastplate of judgment the
Urim and the Thummim; and they shall be
upon Aaron's heart, when he goeth in
before the Lord: and Aaron shall bear the
judgment of the children of Israel upon his
heart before the Lord continually"; in
Leviticus 8:8, we read, "and he put the
breastplate upon him: also he put in the
breastplate the Urim and Thummim."
This method of divining the will of God
became obsolete by 850 B.C. However, it
brings up several interesting questions that
can be answered using elementary probability.
The most obvious is, What is the probability
of getting a "yes" answer to a given
question? The probability is calculated
easily with classical probability and the
combinations formula for counting. The
experiment consists of selecting three stones
at random from a collection of size six. This
can be done in 6C3 = (6!) / (3!3!) = 20 ways.
To get a "yes" answer, all three
stones must be white. This result can
happen in only one way. Thus, the probability
of obtaining a "yes" answer to a particular
question on a given day is 1/20. Due to
the symmetry of the system, the probability
of a "no" answer is also 1/20; the probability that God would not respond is 18/20.