Saturday, 16 July 2011

Large Numbers and Premonitory Dreams

Ever heard of the saying "If at once you don't succeed, then try, try, try again."? Have you ever thought of approaching the problem mathematically? The idea just popped into my head today during math class. So now I shall play the role of a theoretical Mythbuster and explore the validity of age old adages using a a purely theoretical approach.

First let's simplify the problem. Let us assume that the probability of you failing horribly at something (and succeeding at it) remains constant and does not depend on whether or not you failed horribly in your previous attempt. We need to find the minimum number of repetitions of a particular task you need to carry out before the probability of success is 99%.

Mathematically this is a trivial problem which can be solved using a binomial distribution. If $X$ is the Random Variable that represents the number of successes, then $X\sim B(n,p)$ where $n$ is the number of repetitions and p is the probability of success. Let us assume for the purposes of this discussion that you are such an incredible moron at doing this task that your probability of success is only on in hundred or 0.01. We want to find n when the probability of succeeding at least once is 99%. So doing the math,
$X \sim B(n,0.01); P(X\geq 1) = 1-P(X=0) >0.99$

$n>\frac{\log 0.01}{\log 0.99}\Longrightarrow n>458.2$

Repeating the task over 458 times ought to do the trick! So this old proverb does have a mathematical basis!

Premonitory Dreams

So what does all this have to do with Premonitory dreams? As it turns out a similar concept can explain in a very simple and elegant manner why people have premonitory dreams.

There are thousands of reports every year of people dreaming about the death of a friend or close relative days before it happens. The superstitious always think of this as "evidence" for the existence of the supernatural. But despite the fact that science cannot explain individual premonitory dreams, a very simple explanation exists if you consider the population as a whole. Read this extract from an article by Michael Shermer.

We can employ a similar back-of-the-envelope calculation to explain death premonition dreams. The average person has about five dreams a night, or 1,825 dreams a year. If we remember only a tenth of our dreams, then we recall 182.5 dreams a year. There are 300 million Americans, who thus produce 54.7 billion remembered dreams a year. Sociologists tell us that each of us knows about 150 people fairly well, thus producing a social-network grid of 45 billion personal relationship connections. With an annual death rate of 2.4 million Americans, it is inevitable that some of those 54.7 billion remembered dreams will be about some of these 2.4 million deaths among the 300 million Americans and their 45 billion relationship connections. In fact, it would be a miracle if some death premonition dreams did not happen to come true! -  Michael Shermer,September 3, 2008. Why Our Brains Do Not Intuitively Grasp Probabilities. Scientific American, Retrieved from 
As we can see from this article, our brain is not very good at intuitively grasping probabilities, especially when the numbers involved are disproportionately small. This results in the intolerably widespread delusion that premonitory dreams are a result of supernatural influences. It doesn't help that most people are prone to what is called a confirmation bias. When people believe in something they tend to look for events or pieces of evidence that confirm their belief and tend to "filter out" anything and everything that contradicts their beliefs. All those who claim to be psychics take advantage of this cognitive bias. Take tarot card readers as an example. People always remember the time the street psychic predicted their promotion of pay raise but always forget the times they failed to predict anything substantial. In fact if you measured the success rate of psychic predictions you'd probably find that they are not much better than what you would expect from someone who was guessing intelligently. So in the future if you meet somebody who claims that premonitory dreams are proof of the supernatural you can gleefully prove them wrong!


Srikanth said...

True but every time I use this argument there are crazy morons who argue "But what if probability theory is wrong..". Somebody eliminate / educate these people!!

ashwin said...

Haha Probability theory is wrong!! That's a good one!

ashwin said...

Just to make sure, are you able to see the mathematical equations I'm writing or do you just see lines of meaningless code?

Srikanth said...

I see equations alright. Seriously you don't know how many times I get the X theory could be wrong argument. They all start out the same way- Human observation is flawed (Axiom 1)
Theories are developed to explain human observation (Axiom 2)
By implication theories are used to explain the flawed (theorem 1)
By consistency only true statements must be explainable.
Thus by implication: Human theories are wrong (theorem 2)

Anonymous said...

It is not a busted myth if your definition of premonitory dream isn't right.

Dreaming that someone dies and that person actually dies isn't a premonitory dream. Those who have those dreams call it that because they don't know better. They do not know what premonitory dreams actually are.

Premonitory dreams are seeing very specific and exact events unfold while asleep and the events unfold exactly the same way as they did in the dream, the precision being down to the exact words people said, their tone, their demeanor, things like objects falling at the exact same time they did in the dream, etc.

Dreaming that grandpa died in an accident and him dying of a heart attack 3 days later is NOT a premonitory dream. This kind of dream can be dismissed as the subconscious being able to see things the conscious cannot, which is what dream interpretation relies on.

The best example I can give of an actual premonitory dream is a dream a friend of mine had when we were in high school. We were sitting in history class. I turned around and said something to her about being able to go snowboarding next week end, which she mouthed as I said it. She then said "I've dreamed this before. Watch this. The teacher is going to tell us 'Take your book and open it on page 37'".
And so he did.

I've had premonitory dreams of my own, though none being such good examples, since they were never so long that I could start predicting things before the events depicted in the dream were over. However, because I paid close attention to my dreams back then, I could remember thinking of those dreams and trying to interpret them after waking up. So the whole "Oh, your brain is having a time perception glitch and is making up the memory of a dream so it will make sense blah blah" yeah, don't think so.

You're going to need some wacky statistics to be able to make such specific chains of events statistically insignificant.