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 thummin."
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 thummin were used. The sacred container used for divining urim and thummin 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 Thummin; 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 Thummin."
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.