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Before Christmas, NZ Police announced they would be enforcing a zero tolerance on exceeding the speed limit in December/January. Previously there was a 10kph grace unless a driver was obviously driving dangerously. The rationale would be that it would reduce the holiday road toll (deaths through motor vehicle accidents).

Unfortunately the road toll doubled over the previous year.

I'm aware that this could be an anomaly and is subject to many factors, however, is there a related psychological principal (perhaps something around 'suggestion') that states that if a person or population is made aware of a probability of an event, that the probability of that event happening is increased?

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    $\begingroup$ Relevant xkcd $\endgroup$ – Artem Kaznatcheev Jan 12 '15 at 5:02
  • $\begingroup$ Self-fulfilling prophecy? I believe you have a very poor example to describe this however. $\endgroup$ – theMayer Jan 13 '15 at 22:54
  • $\begingroup$ You compare zero tolerance enforced during December and January, that is only one month of last year (December), with the road toll over the whole year. I don't see how the zero tolerance during December could affect the death toll during the eleven months preceding it. $\endgroup$ – user3116 Jan 14 '15 at 12:33
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I believe what you are describing is known as self-fulfilling prophecy and has been well-studied.

Self-fulfilling prophecy are effects in behavioral confirmation effect, in which behavior, influenced by expectations, causes those expectations to come true. It is complementary to the self-defeating prophecy.

It is due to an uncontrolled, positive feedback loop between the expectations of a group of people and their behavior. It applies to a wide variety of situations, such as market panics and bank runs, to expectations of an individual's position in society.

In fact, your example describes a self-defeating prophecy.

A self-defeating prophecy can be the result of rebellion to the prediction. If the audience of a prediction has an interest in seeing it falsified, and its fulfillment depends on their actions or inaction, their actions upon hearing it will make the prediction less plausible.

In essence, the people in your example didn't want the speed limit buffer to go away (in effect, making the speed limit lower) so the theory states that they purposely (if subconsciously) increased the rate at which they crashed to disprove the government's theory that lower speed would prevent crashes.

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By definition, if the probability of an event increases, the probability of the event "happening" increases. Is not an event something that happens?

But I think I know what you're trying to get at. Here's my analysis.

First, let E be the event in question. I think you are trying to express something to the tune of:

P(E) < P(E|P(E))

Taken at face value, P(E) is the probability of the event. P(E|P(E)) is the probability of the event given that we know the probability of the event. This is a situation that is somewhat analogous to stock market dynamics - the more certain the market is that a stock will reach a low price before a certain time, the more they will sell, thus increasing the probability that the low price will be reached before the specified time.

The problem with this formalism is that P(E) is not a random variable. It is a number. E is a random variable. It is therefore associated with some distribution. Perhaps the event has a 30% chance of happening and a 70% chance of not happening. Then drawing values from E will take on one value 70% of the time and another value 30% of the time (however you want to represent happening and not happening - 0 and 1 are always nice).

Now let's suppose P(E) was a random variable. Then the above equation is not expressable. P(E) is not subject to the < operator because it does not represent a single value.

Therefore what I think you are looking for is a bit more Baysian.

Suppose there is a concept Ev(E) which is the evidence that E will occur. This is a proposition, not a random variable. As Ev(E) goes to infinity, P(E) goes to 1. Let P(E) = 1/2 represents the proposition that the you have equal evidence for and against the event happening. You have a certain amount of evidence for it, as represented by Ev(E), and your evidence determines the state of your belief. What you are suggesting is that knowing the value of Ev(E) increases it. Let's make that another proposition:

K = 'I know the value of Ev(E)'

Now consider the statement

Ev(E) < Ev(E|K)

Now,

P(E) < P(E|K). 

What I think you are asking about is really a statement about the change in the amount of evidence you have and its effect on your belief. In the Baysian framework, probabilities are equivalent to beliefs, which is why concepts like this can be addressed a bit more... naturally.

To answer your question, the name for

"awareness of the probability of an event increasing the probability of it   
 happening" 

is equivalent to

"awareness of your belief of an event (happening) increasing your belief of that it 
 will happen"

This phenomenon is called:

 Second guessing your beliefs. 

Or more accurately in this case, as you explicity use the word increasing:

 Becoming more confident in your beliefs.

Perhaps the take away here is that probabilies are not real. They require infinite data to construct and would only be "true: if you could roll back time to collect the exact same data again an again. And if you believe in a deterministic world, then you wouldn't need a probability in the first place. You'd just know. If you believe in a random world and you have the power to roll it back over and over again, and then relive the situation, you get to choose the probability.

That silliness is one of the reasons why probabilities are best understood as represent a state of belief. And if your awareness of your own belief changes it, well that's just questioning yourself and thus reassessing your beliefs.

If you believe in a true probability of an event happening, then your awareness of that probability could not change it, as your awareness is already factored into the true probability of that event happening.

If you're feeling mathy, give this a try: Evidence

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  • $\begingroup$ Thanks for answering this, Doug, but it doesn't get to the root of the question. This isn't about becoming more confident in your beliefs. It's more like the equivalent of 'thoughts create reality' or a 'self-fulfilling prophecy'. I.e. is there a phenomenon in psychology whereby (in this example) if a population is told repeatedly that speeding kills, that it increases the death rate through speeding through an increased belief in the statement's validity, regardless of the probability beforehand? $\endgroup$ – Darren Jan 12 '15 at 22:57
  • $\begingroup$ In my answer, I covered this situaion in markets, which is a more common example. "it increases the death rate through speeding through an increased belief in the statement's validity." It is a reassessment of the beliefs of drivers that changes the frequency of death. Suppose you only told one person that probability speeding kills. Now their probability of dying goes up because they've reassessed their beliefs. They're more likely to die because they believe they will. $\endgroup$ – Doug Jan 13 '15 at 22:49
  • $\begingroup$ I like the term self-edifying truths, but that seems to have a Christian connotation $\endgroup$ – Doug Jan 13 '15 at 22:54
  • $\begingroup$ The term I was looking for was self-fulfilling prophecy from below. $\endgroup$ – Doug Jan 14 '15 at 2:21

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