God plays with loaded dice

I. Introduction

Last Spring, Vern Poythress published Chance and the Sovereignty of God. It's an outstanding treatment. I've been planning to do a post on it, but I was waiting for the ebook edition to come out, because it's easier to quote from the ebook:

http://www.frame-poythress.org/wp-content/uploads/2014/07/ChancePoythress.pdf

Although he doesn't use the terminology, his Scriptural illustrations are textbook examples of coincidence miracles. Likewise, his analysis of chance and probability is useful for unpacking the nature of a coincidence miracle, as well as supplying criteria for the identification of coincidence miracles. Before I quote from his book, let's review some preliminaries.

Traditionally, systematic theology distinguishes between miracles and ordinary providence. A miracle is classically defined as an event that bypasses natural processes. By contrast, ordinary providence employs natural mechanisms. To take a comparison:

i) The development of an acorn into an oak is providential. The acorn has the innate information necessary to turn into an oak. That development follows a continuous process of gestation. 

ii) Take a miracle like turning a stick into a snake (Exod 4). That's naturally impossible. There is no natural mechanism to account for that. 

iii) However, there's a third class of events that overlaps providence and miracle. Suppose a guy dies in an elevator mishap. The elevator suddenly plunges 50 stories, crashing in the basement. 

Normally, we'd consider that a tragic accident, due to a mechanical malfunction. But suppose the victim was an investigative reporter who was about to publish a story that would bring down the president. In that event, we suspect the elevator mishap was a "planned accident" rather than a freak accident. 

Ordinary providence is like a machine that's programmed to do something. It always does and only does what it was programmed to do. Like invariable chemical reactions. 

Compare an assembly line using human workers with robotics. Robots can be programmed to perform some of the same tasks which humans used to do. Although robots are unintelligent, they can perform tasks which require intelligence because they were designed by intelligent engineers who programmed them to perform that task. 

In Scripture, some events are "natural" events in the sense that the outcome is the result of natural means. Yet the outcome is too selective to be the result of blind physical causes. The outcome reflects special guidance. 

Many answered prayers are coincidence miracles. God often answers prayers through natural means. Yet it's not something that would happen if nature was left to operate on its own accord. The result is too discriminating. God coordinated causally independent chains of events to converge at just the right time and place to benefit the Christian.  

The next two sections are verbatim excerpts from the book. 

II. Coincidence miracles

What about seemingly random events? Does God control them?

THE FLIGHT OF AN ARROW

First Kings 22 contains a striking case. Micaiah, speaking as a prophet of the Lord, predicts that Ahab, the king of Israel, will fall in battle at Ramoth-gilead (1 Kings 22:20–22). Ahab disguises himself in battle to avoid being a special target for enemy attack (v. 30). But God’s plan cannot be thwarted. The narrative describes the crucial event:

But a certain man drew his bow at random and struck the king of Israel between the scale armor and the breastplate. Therefore he [the king] said to the driver of his chariot, “Turn around and carry me out of the battle, for I am wounded.” (v. 34)

“A certain man drew his bow at random.” That is, he was not aiming at any particular target. An alternative translation would be that he drew his bow “in his innocence” (ESV marginal reading). The alternative translation might mean that the man shot at Ahab, but he did not know who it was (he was “innocent” of knowing it was the king). Whichever interpretation we take of this detail, we should notice that the arrow struck in just the right place. Ahab was dressed in armor. If the arrow had struck Ahab’s breastplate, it might have simply bounced off. If it had struck his scale armor, it would not have wounded him. But there happened to be a small space between the scale armor and the breastplate. Perhaps for just a moment Ahab turned or bent in such a way that a thin opening appeared. The arrow went right in, exactly in the right spot. It wounded him fatally. He died the same day (1 Kings 22:35), just as God had said.

God showed that day that he was in charge of seemingly random events. He controlled when the man drew his bow. He controlled the direction of his aim. He controlled the moment the arrow was released. He controlled the flight of the arrow. He controlled the way Ahab’s armor was put on earlier in the day, and the position that Ahab took as the arrow came nearer. He controlled the arrow as it struck in just the right spot and went in deep enough to produce fatal damage to organs. He brought Ahab to his death.

Lest we feel too sorry for Ahab, we should remind ourselves that he was a wicked king (1 Kings 21:25–26). Moreover, by going into battle he directly disobeyed the warning that Micaiah the prophet gave in God’s name. It was an act of arrogance and disobedience to God. God, who is a God of justice, executed righteous judgment on Ahab. From this judgment we should learn to revere God and honor him.

Ahab’s death was an event of special significance. It had been prophesied beforehand, and Ahab himself was a special person. He was the king of Israel, a prominent leader, a key person in connection with the history of God’s people in the northern kingdom of Israel. But the event illustrates a general principle: God controls seemingly random events. A single out- standing event, like the arrow flying toward Ahab, has not been narrated as an exception but rather as a particularly weighty instance of the general principle, which the Bible articulates in passages where it teaches God’s universal control.

COINCIDENCES

We can find other events in the Bible where the outcome depends on an apparent coincidence or happenstance.

In Genesis 24, Rebekah, who belonged to the clan of Abraham’s relatives, happened to come out to the well just after Abraham’s servant arrived. The servant was praying and waiting, looking for a wife for Abraham’s son Isaac (Gen. 24:15). The fact that Rebekah came out at just the right time was clearly God’s answer to the servant’s prayer. Rebekah later married Isaac and bore Jacob, an ancestor of Jesus Christ.

Years later Rachel, who belonged to the same clan, happened to come out to a well just after Jacob arrived (Gen. 29:6). Jacob met her, fell in love with her, and married her. She became the mother of Joseph, whom God later raised up to preserve the whole family of Jacob during a seven-year famine (Genesis 41–46). When God provided Rachel for Jacob, he was fulfilling his promise that he would take care of Jacob and bring him back to Canaan (28:15). Moreover, he was fulfilling his long-range promise that he would bless the descendants of Abraham (vv. 13–14).

In the life of Joseph, after Joseph’s brothers had thrown him into a pit, a caravan of Ishmaelites happened to go by, traveling on their way to Egypt (Gen. 37:25). The brothers sold Joseph to the Ishmaelites. They in turn happened to sell Joseph to Potiphar, “an officer of Pharaoh” (v. 36). Joseph’s experiences were grim, but they were moving him toward the new position that he would eventually assume in Egypt.

False accusation by the wife of Potiphar led to Joseph being thrown into prison (Gen. 39:20). Pharaoh happened to get angry with his chief cupbearer and his chief baker, and they happened to get thrown into the prison where Joseph now had a position of responsibility (40:1–4). While they were lying in prison, both the cupbearer and the baker happened to have special dreams. Joseph’s interpretation of their dreams led to his later opportunity to interpret Pharaoh’s dreams (Genesis 41). These events led to the fulfillment of the earlier prophetic dreams that God had given to Joseph in his youth (37:5–10; 42:9).

After Moses was born, his mother put him in a basket made of bulrushes and placed it among the reeds by the Nile. The daughter of Pharaoh happened to come down to the river and happened to notice it. When she opened it, the baby happened to cry. The daughter of Pharaoh took pity and adopted Moses as her own son (Ex. 2:3–10). As a result, Moses was protected from the death sentence on Hebrew male children (1:16, 22), and he “was instructed in all the wisdom of the Egyptians” (Acts 7:22). So God worked out his plan, according to which Moses would eventually deliver the Israelites from Egypt.

Joshua sent two spies to Jericho. Out of all the possibilities, they happened to go to the house of Rahab the prostitute (Josh. 2:1). Rahab hid the spies and made an agreement with them (vv. 4, 12–14). Consequently, she and her relatives were preserved when the city of Jericho was destroyed (6:17, 25). Rahab then became an ancestor of Jesus (Matt. 1:5).

Ruth “happened to come to the part of the field belonging to Boaz” (Ruth 2:3). Boaz noticed Ruth, and then a series of events led to Boaz marrying Ruth, who became an ancestor of Jesus (Ruth 4:21–22; Matt. 1:5).

During the life of David, we read the following account of what happened in the wilderness of Maon:

As Saul and his men were closing in on David and his men to capture them, a messenger came to Saul, saying, “Hurry and come, for the Philistines have made a raid against the land.” So Saul returned from pursuing after David and went against the Philistines (1 Sam. 23:26–28).

David narrowly escaped being killed, because the Philistines happened to conduct a raid at a particular time, and the messenger happened to reach Saul when he did. If nothing had happened to interfere with Saul’s pursuit, he might have succeeded in killing David. The death of David would have cut off the line of descendants leading to Jesus (Matt. 1:1, 6).

When Absalom engineered his revolt against David’s rule, a messenger happened to come to David, saying, “The hearts of the men of Israel have gone after Absalom” (2 Sam. 15:13). David immediately fled Jerusalem, where otherwise he would have been killed. During David’s flight, Hushai the Archite happened to come to meet him, “with his coat torn and dirt on his head” (v. 32). David told Hushai to go back to Jerusalem, pretend to support Absalom, and defeat the counsel of Ahithophel (v. 34). As a result, Hushai was able to persuade Absalom not to follow Ahithophel’s counsel for battle, and Absalom died in the battle that eventually took place (18:14–15). Thus, happenstances contributed to David’s survival.

When Benhadad the king of Syria was besieging Samaria, the city was starving. Elisha predicted that the next day the city of Samaria would have flour and barley (2 Kings 7:1). The captain standing by expressed disbelief, and then Elisha predicted that he would “see it . . . but . . . not eat of it” (v. 2). The next day the captain happened to be trampled by the people who were rushing out the gate toward the food (v. 17). “He died, as the man of God had said” (v. 17), seeing the food but not living to partake of it. His death was a fulfillment of God’s prophecy.

When Athaliah was about to usurp the throne of Judah, she undertook to destroy all the descendants in the Davidic family. Jehosheba happened to be there, and she took Joash the son of Ahaziah and hid him away (2 Kings 11:2). So the line of the Davidic family was preserved, which had to be the case if the Messiah was to come from the line of David, as God had promised. Joash was an ancestor of Jesus Christ.

During the reign of king Josiah, the priests happened to find the Book of the Law as they were repairing the temple precincts (2 Kings 22:8). Josiah had it read to him, and so he was energized to inaugurate a spiritual reform.

The story of Esther contains further happenstances. Esther happened to be among the young women taken into the king’s palace (Est. 2:8). She happened to be chosen to be the new queen (v. 17). Mordecai happened to find out about Bigthan and Teresh’s plot against the king (v. 22), and Mordecai’s name then happened to be included in the king’s chronicles (v. 23). The night before Haman planned to hang Mordecai, the king happened not to be able to sleep (6:1). He asked for an assistant to read from the chronicles, and he happened to read the part where Mordecai had uncovered the plot against the king (vv. 1–2). Haman happened to be entering the king’s court at just that moment (v. 4). A whole series of happenstances worked together to lead to Haman’s being hanged, the Jews being rescued, and Mordecai being honored.

The book of Jonah also contains events that worked together. The Lord sent the storm at sea (Jonah 1:4). When the sailors cast lots in order to iden- tify the guilty person, “the lot fell on Jonah” (v. 7). The Lord appointed the fish that swallowed Jonah (v. 17). The Lord also appointed the plant that grew up (4:6), the worm that attacked the plant (v. 7), and then the blazing of the sun and the “scorching east wind” (v. 8).

Zechariah the priest, the husband of Elizabeth, happened to be chosen by lot to burn incense in the temple (Luke 1:9). The time was just right, shortly before the conception of John the Baptist and the coming of Jesus (vv. 24–38).

When Dorcas died in Joppa, Peter happened to be nearby in Lydda (Acts 9:32, 38). The disciples in Joppa happened to hear that he was there. So they sent for Peter, and as a result Dorcas was raised back to life.

While Paul the apostle was in prison, the son of Paul’s sister happened to hear about the Jewish plot to kill Paul (Acts 23:16). He passed the news on to the Roman leader, the tribune, who had his soldiers take Paul to Caesarea. Paul was saved from being killed because of a happenstance.

We could multiply instances of this kind. The storm and the fish that the Lord sent to Jonah might be considered miraculous, but for the most part we have focused on incidents where a bystander may not have noticed anything extraordinary. In each case, the narrative as a whole shows that God was accomplishing his purposes (chap. 3).

We can confirm the point about God’s control over apparently random events with another case, namely the disasters that befell Job.

DISASTERS IN THE BOOK OF JOB

Job 1 describes several disasters. The key passage is worth quoting in full:

Now there was a day when his [Job’s] sons and daughters were eating and drinking wine in their oldest brother’s house, and there came a messenger to Job and said, “The oxen were plowing and the donkeys feeding beside them, and the Sabeans fell upon them and took them and struck down the servants with the edge of the sword, and I alone have escaped to tell you.” While he was yet speaking, there came another and said, “The fire of God fell from heaven and burned up the sheep and the servants and consumed them, and I alone have escaped to tell you.” While he was yet speaking, there came another and said, “The Chaldeans formed three groups and made a raid on the camels and took them and struck down the servants with the edge of the sword, and I alone have escaped to tell you.” While he was yet speaking, there came another and said, “Your sons and daughters were eating and drinking wine in their oldest brother’s house, and behold, a great wind came across the wilderness and struck the four corners of the house, and it fell upon the young people, and they are dead, and I alone have escaped to tell you.”Then Job arose and tore his robe and shaved his head and fell on the ground and worshiped. And he said, “Naked I came from my mother’s womb, and naked shall I return. The Lord gave, and the Lord has taken away; blessed be the name of the Lord.”
In all this Job did not sin or charge God with wrong. (Job 1:13–22)

Some of these disasters seem to be random. For one thing, how come they all happened on the same day? That in itself seems unlikely, because they are not causally connected to one another. One of the disasters was that “the fire of God fell from heaven” (Job 1:16). When and where it would fall was totally unpredictable. Why did it fall when it did on Job’s sheep and servants, and not elsewhere? How was it that “a great wind” came (v. 19), and why did it hit the house and not elsewhere, and why did it hit at the moment when Job’s sons and daughters were inside the house?

Job was faced with a series of seemingly random events. He was emotionally devastated by the losses. But how did he deal with the question of why? Did he think, “Well, things just happen by chance because the world has chance in it”? No, he saw the hand of God: “The Lord gave, and the Lord has taken away” (1:21).

A consistent deist would have to say, “It was all part of the clockwork.” Deism might lead to the conclusion that God created the world with both order and randomness. According to deistic thinking, the randomness just has to be accepted. God is not responsible for disasters, because he has walked away from the clock that he made. Other people might still want God to be responsible for the good things and the blessings that come to us. But they cannot stomach the idea that he was responsible for a disaster like Job’s. They would say that they want to protect the goodness of God.

Yes, the Bible does teach that God is good and does good (Ps. 86:5; 100:5; 107:1; 119:68). But it flatly contradicts those who want to “protect” him by removing his control over disasters. Job made it clear that he thought God was in control: “The Lord gave, and the Lord has taken away” (Job 1:21). Was Job wrong? From the surrounding narrative in Job 1 we learn that Satan engineered the disasters:

And the Lord said to Satan, “Behold, all that he has is in your hand. Only against him do not stretch out your hand” (Job 1:12).

But Satan did not act without God’s permission (see Job 1:10–11). We see three distinct causes: God, Satan, and human raiders (vv. 15, 17), all acting within the same events. The plans of Satan do not negate the sovereignty of God (ibid. 41-43).

III. Controlled "chance"

The Bible makes it clear by any number of cases that God involves himself in details:

Are not two sparrows sold for a penny? And not one of them will fall to the ground apart from your Father. (Matt. 10:29) 

But even the hairs of your head are all numbered. (Matt. 10:30) 

. . . to bring rain on a land where no man is, on the desert in which there is no man,
to satisfy the waste and desolate land, and to make the ground sprout with grass? (Job 38:26–27) 

Lift up your eyes on high and see: who created these?
He who brings out their host by number, calling them all by name,
by the greatness of his might, and because he is strong in power not one is missing. (Isa. 40:26)

Consider now a classic case of a random event: the roll of dice. When we roll dice, no one can predict what numbers will come up. The result is a matter of pure “chance.” Here is what the Bible says:

The lot is cast into the lap, but its every decision is from the Lord. (Prov. 16:33)

The expression “the lot” designates some kind of random event. It covers a range of possible means. People can roll dice, or flip a coin, or spin a top, or spin a dial with markings on it. Or they may throw down sticks and observe whether they form a pattern of some kind. The fact that the lot “is cast into the lap” suggests in this case something more like dice. Whatever the means used, “its every decision is from the Lord.” “Every decision,” it says, not just some. Every time the dice come up, they come up as the Lord directs. The Lord controls the outcome of this random event.

A skeptic might still claim that Proverbs 16:33 covers only a few “special” events. The proverb envisions primarily a situation where people cast a lot in order to make a decision based on the outcome of the lot. They might have an important religious or political decision to make.

In Joshua 7:14 we see a significant incident where lots are used. Someone in Israel has taken things out of Jericho that were “devoted” to God, which God had claimed for himself and told the people not to take. Joshua then uses lots to find out which tribe and which member of the tribe has done the deed. The outcome of the lots does take place under the Lord’s control, because they find out that Achan is the culprit (Josh. 7:18).

In more pleasant circumstances, in 1 Samuel 10:20–21, the casting of lots singles out Saul the son of Kish as the new king of Israel. A lot also singles out Jonah as the person responsible for the storm at sea (Jonah 1:7). A lot is used by the apostles in Acts 1 to determine whether Joseph called Barsabbas or Matthias should be appointed as an additional apostle, to fill the place left empty by the death Judas Iscariot (Acts 1:23–26). The successor to Judas must be the one whom the Lord has appointed, and the will of the Lord comes to expression when the apostles cast lots. “The lot fell on Matthias, and he was numbered with the eleven apostles” (v. 26). The apostles clearly understand that the outcome for this casting of lots is controlled by the Lord.

We can see a similar kind of thing in modern times when a group of people draw straws or flip a coin to see who goes first. Sometimes the result may be humanly important, if they are risking their lives in a dangerous mission. Sometimes the result may be of small importance, if they are just determining which person plays first in a game.

So, the skeptic wonders, does God’s control over dice or lots take place only when some weighty decision is needed? Or, even more narrowly, does his control apply only to intense religious situations in Israel, such as selecting Achan or Saul or Matthias? Or does God’s control extend to other instances?

The verse in Proverbs 16:33 does not have any qualification. It does not say, “When an important decision has to be made, the decision is from the Lord.” The formulation is a general one: “the lot is cast into the lap.” The natural meaning is, “any lot whatsoever.” It includes the lot cast by the pagan sailors on Jonah’s ship. “Every decision,” not merely a decision once in a while, is “from the Lord.” It is true that the proverb focuses on lots that have some significance, because such lots are the ones in which people are most interested, and where it is most important that they understand the Lord’s control. But the principle is a general one: every lot. Every lot has its outcome determined by the Lord in his sovereignty, and in accord with his eternal plan. We can generalize further: the Lord controls every random event, whether it is deliberately brought about by a human action of rolling dice or flipping coins, or is just a happenstance, like a hair coming out of someone’s head and falling to the ground.

How do we know this? We know this because Proverbs 16:33 is a general principle. It has no qualifications that would limit the power of God over details. The absence of limitation agrees with the verses that we have already seen that teach the complete universality of God’s control:

. . . having been predestined according to the purpose of him who works all things according to the counsel of his will. (Eph. 1:11) 

Who has spoken and it came to pass, unless the Lord has commanded it?
Is it not from the mouth of the Most High that good and bad come? (Lam. 3:37–38) [ibid. 63-67]

We can rely on another regularity, called independence of events or independence of probabilities. Independence is a key idea in the theory of probability, but it takes some explaining. Suppose we have two dice, one white and one red. The probability that the white die will come up 5 is 1/6. The probability that the red die will come up 5 is also 1/6. These truths follow from symmetry and also from the regularities in space and time.

Now picture a situation in which we roll the white die, and it comes up 5. Then we proceed to roll the red die. What is now the probability that it will come up 5, given the extra information that we have, namely, the information that the white die has already come up 5?

The actual answer is that the red die still has a 1/6 probability of coming up 5. Knowing the outcome from the white die does not affect the red die. Its probabilities are still the same as they were before. The technical term for this situation is probabilistic independence. We say that the outcome for the red die is independent of the outcome for the white die. This kind of independence does not occur in our examples about the 75-year-old woman who smokes or exercises regularly. The probability that she will die in the next year is influenced by such extra information. It is not independent of the information. Some kinds of knowledge influence probability estimates, but other kinds of knowledge do not. When one kind of knowledge does not have an influence, we describe the situation as a situation of probabilistic independence. 

The roll of one white die does not affect the outcome of the roll of a red die. The two are independent. Similarly, a previous roll of a white die does not affect the outcome of the next roll of the same die. This independence is an independence in time.

Some people’s intuitions fail them when they think about situations like these. For example, they may imagine that since the white die has already come up 5, a second roll of the same die is less likely to come up 5. They may try to bolster their reasoning by pointing out that the average for a large number of die rolls must work out so that the outcome of 5 is no more frequent than any other outcome. So surely the next roll is a little less likely to come up a 5, in order to “balance” the long-run frequencies of all six outcomes. By similar reasoning, if a single die has come up 5 six times in a row, it is quite a bit less likely to come up 5 again, because it has to balance out the total number of 5s with the totals for the other possible outcomes.

Some people’s intuitions may actually go in the opposite direction. They may think that, after several occurrences of an outcome of 5, the die is more likely to come up 5 because maybe there is a tendency to stick to a pattern that is already in place.

There are indeed situations in ordinary life that show patterns like these. Suppose you go to a Little League game knowing nothing about either team. You watch the pitcher, and the first eight pitches you see are all strikes. Is the next pitch likely to be a strike? Yes. There is a good chance that you are watching a very accurate pitcher, and that he has decided to try to throw a strike every time. You learn from watching that there is a pattern to his pitches. The probability of his throwing a strike is very high, especially when compared to another pitcher with poor accuracy.

Now let us go back to the situation with dice. We have to see that the two dice are more like two pitchers than one. Just because one pitcher is accurate, it does not make another pitcher more accurate. The same is true for the situation where we repeatedly roll a single die. We throw the white die a second time, a third time, and so on. Is it more likely to come up 5? What if it comes up 5 three times in a row? Is it likely to come up 5 on the fourth throw? The answer is no. The fourth throw still has a probability of 1/6 of coming up 5. If it comes up 5 ten times in a row, or a hundred times in a row, the probability of coming up 5 on the next roll is still 1/6. That is what we mean by probabilistic independence.

But we must insert a qualification. The probabilities we are talking about for dice are a priori probabilities. We knew what these probabilities were before we ever starting rolling the dice. But suppose we start for the first time with rolling a die, and it does come up 5 a full 10 times in a row, right after we start. What then? That is a very unusual result, so unusual that we begin to suspect that there is something fishy. Someone has tampered with the die. It looks symmetrical, but maybe it is not. Ah, it feels funny. The face opposite to the 5 seems to be very heavy. What is happening here is that in our assessment of the die we are being influenced by a posteriori probabilities. The actual results of conducting trials, that is, conducting rolls, are so unusual that we look around for some explanation for why the results, that is, the a posteriori samples, differ strongly from the a priori predictions.

Gamblers sometimes get trapped by their feelings or hunches about probabilities. They feel that a particular die or a roulette wheel or other object has mysteriously gotten “stuck” on some pattern, and therefore it is very likely that the pattern will continue. Or, conversely, they notice that 5 has not come up for a long time on the die, so, they feel, it is “time” for it to come up, and the probability of it coming up on the very next roll is higher than it would otherwise be. Are they right? The answer is no. The patterns that the gamblers think that they see are all temporary, ephemeral. Despite the gamblers’ feelings, the outcome of the next roll of the die is just as unpredictable as the very first roll. The probability of coming up 5 is 1/6. This probability is independent of all the previous rolls, as far back as we go.

How do we know that is the case? We are finite; we do not know absolutely. But those who have studied events like repeated coin flips and repeated dice rolls and repeated drawing of cards from well shuffled decks discern a pattern of independence in all these types of events. The pattern is ordained by God in his faithfulness and creativity and love.

We can, in part, understand something of the rationale and the wisdom in this pattern. Each roll of a die is distinct. And each is going to involve minute differences in the initial orientation of the die, and how it first strikes the ground, and so on. Such differences cannot be controlled by human beings. So the spatial symmetry of the die’s faces do suggest, by means of a priori reasoning, that the six distinct outcomes should be equally likely. And since each roll of each die is different in the details of how it starts, there will be no intrinsic correlation between two distinct rolls or two distinct dice. The lack of intrinsic correlation means independence.

This independence contrasts with the intrinsic correlations that we sense do exist in cases where we consider, for example, the relation of smoking or family history to the likelihood of death. Things that happen in the woman’s body earlier in time influence the state of her health. By contrast, the history of a die does not influence the next roll, because the roll starts fresh with slightly different orientation, slightly different rate of spin, and so on (ibid. 191-94).

The casino will soon notice his success. Winning in this way is so un- usual that the casino manager might suspect that the gambler has formed a secret partnership with the employee managing the roulette table, and that together they have found some secret way of manipulating the outcome of the wheel. If the manager can find no explanation of this kind, he will nevertheless ban the gambler from the roulette table beginning on the next day. He cannot afford to do otherwise. If he were to let the gambler continue, he would continue losing money to the one gambler. But in addition, other gamblers would soon notice the “good luck” and begin to imitate his bets, thereby “piling on” and winning money themselves (ibid. 205).