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- SEVEN -
COINCIDENCES AND MIRACLES
Coincidences
The most astonishingly incredible coincidence imaginable would be the complete absence of all coincidences.
- John Allen Paulos (from Beyond Numeracy)
We notice certain things and ignore others. We select and talk about the amazing event, not the ordinary ones. We see coincidences after they happen. We don't see them before they happen.
We underestimate how many opportunities there are for "unlikely" events to happen. Surprises and improbable events happen if they have enough opportunities to happen. There are many ways in which events may be linked together.
Getting 5 tails in a row is certain to happen somewhere, sometime to someone. The chances may be small that an event happens at a particular place, time or to a particular person. But with many places, over long periods of time or with many individuals, the seemingly improbable will happen. As Aristotle says: "It is likely that unlikely things should happen."
Someone flips tails 20 times in a row.
Amazing, isn't it? Seen as an isolated event it may seem unlikely. But with a large enough group to choose from, it is likely that it happens to someone. In a group of 1,048,576 people it happens to someone. In fact, in the U.S. a country with about 280 million people, one in a million chance events happen 280 times a day.
How likely is it that two people share the same birthday?
There are many opportunities for coincidences. For example, in a group of 23 people, the probability is 50.7% that two people share the same birthday. It is rather likely that events like this one happen since there are many ways that 2 people can share an unspecified birthday. Observe that the question is not how likely it is that 2 people share any particular birthday. The question is whether it is likely that 2 people share an unspecified birthday.
How many people have to be present at Mary's birthday dinner so there is more
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than a 50% chance that one of the other guests has the same birthday as Mary? 183, since we now work with the restriction "two people must share a particular birthday."
Scale matters. When the numbers are large enough, improbable things happen. The more available opportunities or the longer the time, unlikely events happen. Ask: What are all possible outcomes and their likelihood? What else could have happened?
Making up causes for chance events
Humans are pattern seeking, storytelling animals. We look for and find patterns in our world and in our lives, then weave narratives around those patterns to bring them to life and give them meaning.
- Michael Shermer (Publisher of Skeptic)
john tosses a single die six times and the result is either (A) 623514 or (B) 666111.
Which alternative showed the correct outcome? Even if (A) looks random and
looks like it has a pattern, in independent chance events, both A and Bare equally likely or unlikely to appear. Random events may not look random. Independent sequences often show order or streaks. For example, during the Second World War, people saw a pattern in the German bomb hits of London, expecting some areas more dangerous than others. But they were randomly distributed over London.
We want to find reasons for all kind of events - random or not. We search for patterns even where none exist. For example, there must be something important that is happening if a particular number comes up again and again. But it is always possible to find patterns and meaning in an event if we actively search for them and selectively pick anything that fits the pattern and ignore everything that doesn't. But we can't predict the pattern in advance.
John rolls five dice and gets five sixes.
The probability that some roll contains any five specified numbers is very small or one in 7,776 (for example, 6x6x6x6x6). When we toss five dice there are precisely 7,776 different combinations of numbers that can appear. Every combination is equally likely and one of them is certain to happen every time we roll five dice. Even if one particular combination (five sixes) is improbable, no single combination is impossible. Any combination that happens is simply one out of a number of equally likely outcomes. It may have been improbable that John rolled five sixes, but not impossible.
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Anything can happen if the number of possibilities is large. People have seen a human face on Mars, faces in rocks, clouds or even in a grilled cheese sandwich. But that is no mystery. Given the large numbers of rocks, clouds, and sandwiches, sooner or later we will find one that looks like a face, even a particular face.
Believing in miracles
Impector Gregory: '1s there any point to which you would wish to draw my attention?" Sherlock Holmes: "To the curious incident of the dog in the night-time. "
Inspector Gregoty: "The dog did nothing in the night-time. "
"That was the curious incident,"remarked Sherlock Holmes.
- Arthur Conan Doyle (from Silver Blaze)
Mary thinks about calling her friend Jill. Suddenly the phone rings and it is Jill.
Is something paranormal going on? No, Mary forgot about all the times Jill didn't call when Mary was thinking about her, or the times someone else called, or the times Jill called but Mary wasn't thinking of her, or the times Jill didn't call when Mary wasn't thinking of her. When Mary thinks about Jill and the phone rings, it registers as an event and something we remember. When Jill doesn't call, it is a non-event. Nothing happened. Nothing is registered and therefore nothing is remembered.
We often pay little or no attention to times when nothing happens. We shouldn't look at past events and find significance in the amazing ones. We need to make comparisons among cases involving no cause or no effect and look at all the other things that might have happened instead.
"The psychic predicted the tornado"
Amazing. It sounds too good to be by chance. What we didn't know was that the psychic predicts a tornado every week. As Marcus Tullius Cicero said: "For who can shoot all day without striking the target occasionally." Often we don't notice the incorrect predictions, only the rare moments when something happens. We forget when they are wrong and only remember when they were right. And many times we want them to be right, so we hear what we want to hear and fill in the blanks.
"The art of prophecy is very difficult, especially with respect to the future", wrote Mark Twain. This is why it's important to remain skeptical of "future tellers." Their right guesses are highly publicized but not all their wrong guesses. Like Harvard Professor Theodore Levitt said: "It's easy to be a prophet. You make twenty-five predictions and the ones that come true are the ones you talk about." Michel de Montaigne adds: "Besides, nobody keeps a record of their erroneous prophecies since they are infinite and everyday."
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The future-tellers predictions are always far enough in the future that they never have to face the consequences when they're wrong. Or they make their forecasts so general they can apply to anyone or to any outcome so they can't be proved wrong.
"There is no evidence that ghosts don't exist. "
Some things can't be proven false. The fact that there is no evidence against ghosts isn't the same as confirming evidence that there are ghosts. What is true depends on the amount of evidence supporting it, not by the lack of evidence against it.
Mary comes home after school and tells John: "My friend Alice witnessed a miracle. " The 18th Century Scottish Philosopher David Hume suggested a test to analyze claims of miraculous events: "No testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous than the fact which it endeavors to establish."
Hume suggests the following test: If the opposite of a given statement is more
likely, the statement is probably false. Thus, isn't it more likely that the opposite, "Alice didn't witness a miracle" is true? Not because miracles are impossible but because the alternative explanation of illusion is more probable. How many things that are impossible must happen fo
r a miracle to be true?
The German poet Johann Wolfgang von Goethe said: "Mysteries are not necessarily miracles." That an event can't be explained doesn't mean it is a miracle. No theory can explain everything. As Michael Shermer says, "My analogy is that the L.A.P.D. [Los Angeles Police Department] can solve, say, 90 percent of annual homicides. Are we to assume that the other 10 percent have supernatural or paranormal causes? No, of course not, because we all understand that the police cannot solve all murder mysteries."
Bertrand Russell said in A History of Western Philosophy: "Uncertainty, in the presence of vivid hopes and fears, is painful, but must be endured if we wish to live without the support of comforting fairy tales."
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- EIGHT -
RELIABILITY OF CASE EVIDENCE
Prior probabilities
At the dinner table john tells Mary: "One of our employees got caught today for stealing, but she said she never did it before and will never do it again. "
How likely is it that she never did it before? Look at the prior probability of stealing, considering for example, the base rate or how typical or representative an event or attribute is.
Charles Munger tells us how John should think:
If you're going to catch 10 embezzlements a year, what are the chances that any one of them - applying what Tversky and Kahneman called base rate information - will be somebody who only did it this once? And the people who have done it before and are going to do it again, what are they all going to say? Well in the history of the ... Company they always say, "I never did it before, and I'm never going to do it again." And we cashier them. It would be evil not to, because terrible behavior spreads.
Does eyewitness identification or DNA evidence mean that a person is guilty? Does a positive medical test mean that a person has a disease?
In the eighteenth century, the English mathematician and minister Thomas Bayes laid a foundation for a method of evaluating evidence. The French mathematician Pierre Simon de Laplace brought the method to its modern form. Bayes' Theorem makes it possible to update the prior probability of an outcome in light of new evidence. It is easier to use if we change probability formats into frequency formats. Let's use Bayes's Theorem with a modified version of the classic cab problem,
originally developed by the Psychologists Daniel Kahneman and Amos Tversky.
john testifies in court: '1 witnessed the accident and the cab involved was green. " John's vision has been reliably tested and the tests establish that he can identify the color green correctly 8 out of 10 times. John said "green" in 8 out of 10 cases when something was green and said "green" 2 times out of 10 when something was blue. This means that John misidentified the color 2 out of 10 times.
How trustworthy is John as an eyewitness? A witness testimony always
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contains a degree of uncertainty. Also remember that the reliability of any observation not only depends on the reliability of the observer - even if John has good eyes - but also on how likely his observation is true given prior probabilities. First we ask: What is the prior probability ofoutcome- how probable is an event prior to considering the new evidence? How probable is it that a green cab was involved in an accident before we consider John's evidence? Assume that the relative frequency (the proportion of cabs of a certain color in a particular population at a specific point in time) of blue and green cabs gives us information about the prior probability of involvement in the accident. What was the proportion of blue and green cabs out of all cabs at the time of the accident? Assume there were a total of 100 cabs in town. 90 blue and 10 green. This means that the prior probability the
cab involved was green is 10% (10 green cabs out of 100).
What is the (posterior - after considering case evidence) probability the cab was green given that John said it was green?
John says "green"
Given green
8 cabs (IO x 0.8)
Given blue 18 (90 X 0.2)
Total 26
If 10 out of 100 cabs are green and John is right 8 out of 10 times, then he identifies 8 cabs as green. IfJohn says green cab and it is not a green cab then he is likely to identify 18 of the 80 blue cabs as green. Out of a total of 26 cabs John has identified as green, only 8 are green. This means that the likelihood that the cab was green given John's testimony that "the cab was green" is 31 % (8/26). It seems the cab involved is more likely to have been blue.
Before John testified the prior probability was only 10% that the cab involved was green. When he testified "green" the probability rose to 31%.
"Based on 50 years of accidents involving cabs, given the same proportion of color, 3 out of 4 times the cab involved was green. "
Independent of the frequency of blue and green cabs, the appropriate prior probability may have been past evidence of accidents. What we want is the correct evidence that is representative for what was likely to happen before we considered the new evidence.
When we get new representative evidence, we must update the prior probability. Ask: What has happened in similar cases in the past? Are there any reasons this probability should be revised? Have the circumstances or the environment changed? The more uncertainty that surrounds a specific case, the more emphasis we must put on the prior probability.
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How strong is the evidence?
One factor when evaluating evidence is the coincidental or random match probability. It answers the question: What is the probability that a person other than the suspect, randomly selected, will match a certain profile? For example, when evaluating DNA evidence a random match happens when two different people share the same DNA profile.
After five days of searching, the police found the missing woman strangled to death. john's brother, Bill, is on trial for her murder.
There is a DNA profile match. The forensic evidence against Bill is blood and tissue samples taken from the crime scene, which match Bill's. Either Bill left the evidence or someone else did.
What is the probability of a coincidental match? How likely is it that a match occurs between the DNA profile found at the scene of the murder and a randomly chosen person? How likely is it that Bill's profile matches the profile of the person who did leave the evidence at the crime scene? How rare is this profile? The rarer the profile, the lower the probability that Bill's matches only by chance. The prosecution's medical expert witness estimates (estimation of the frequency of the profile in the most appropriate comparison population) the probability that there would be such a match if Bill was innocent and the match was just a coincidence as only 1 in 20,000. This means that out of every 20,000 individuals, only one will have the same DNA profile as the one found at the murder scene. The prosecutor argues: "There is only a one in 20,000 chance that someone other than Bill would by chance have the same profile as the one found on the murder scene. The probability is therefore only 1 in 20,000 that someone other than Bill left the evidence. "The figure had a dramatic impact on the media and
the jury. Bill was found guilty and given a life sentence.
Where did justice go wrong? The prosecutor confused two probabilities. The probability that Bill is innocent given a match is not the same as the probability of a match given Bill is innocent. The prosecutor should have said: "The probability is one in 20,000 that some person other than Bill would leave the same blood and tissue as that found on the crime scene. "
The jury also needs to consider prior (before the evaluation of the forensic evidence is considered) probabilities of guilt. The probability that Bill is the murderer can't only be calculated from the forensic evidence alone. Other evidence needs to be considered. The significance of forensic evidence always depends on other evidence. What other data do the police have? What else is known about Bill? Does he have an alibi? Was he near the crime scene? Each piece
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of evidence must be considered together, and not in isolation. There may also have been a non-criminal explanation of why Bill left the b
lood and tissue samples.
Based on evidence prior to considering the forensic evidence the jury estimates there is a 10% probability that Bill is the source of the forensic evidence (90% that he is not and thus innocent). The probability of a match given Bill is guilty is 1 (sensitivity 100% i.e. no false negatives) because if Bill is the source of the forensic evidence and the laboratory test is accurate, his DNA profile will match. Combining this with the random match probability of one chance in 20,000 (that his DNA profile showed up at the crime scene just by chance) gives a posterior probability that Bill is the source of the forensic evidence of 99.96% (0.1/0.100045).
Match
Given guilty Given innocent
0.1 (10% X 100%) 0.000045 (90% X 1/20,000)
Total 0.100045
One way of determining the prior probability is by asking: What is the population from which the murderer could have come? We need to know the appropriate comparison population to estimate this number. The murder has taken place in a city of 500,000 men. Assume any man in the city could have committed the crime. One of them is the murderer. Of the 499,999 people innocent, we can expect about 25 coincidental DNA matches. This means there are 26 men (25 + the murderer) who could have committed the crime. Since Bill is one of these 26, the probability that he is guilty given the forensic evidence, is only 3.8% (1/26).
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