Improbable things happen
Armondikov (Talk | contribs) m (→Paul the Octopus) |
m (Robot: Changing Category:False arguments) |
||
| Line 56: | Line 56: | ||
==Footnotes== | ==Footnotes== | ||
<references/> | <references/> | ||
| + | |||
[[Category:Statistics]] | [[Category:Statistics]] | ||
| − | [[Category: | + | [[Category:Fallacious arguments]] |
[[Category:Logic]] | [[Category:Logic]] | ||
Revision as of 07:06, 7 September 2011
| Part of the series on |
| Key articles |
| General logic |
| Bad logic |
Improbable things happen all the time.
Creationists, anti-evolutionists and all manner of non-rationalists like to disparage their opponents or bolster their own arguments by pointing out the improbability of something happening. But, improbable things happen all the time, because the "improbability" is an illusion based on our preconceptions, and not a statistical truth.
In short, "improbability" does not imply "impossibility".
According to the ergodic hypothesis, provided the ultimate laws of physics possess certain reasonably possible (but unproven) characteristics, then every event with non-zero probability, however small, will eventually occur.
In an infinite universe, events with zero probability can happen - although, in order for them to have zero probability, they can only occur a finite number of times.
Contents |
The lottery
Possibly the simplest example is a lottery. These often have incredible odds that seem impossible to beat, but indeed someone (almost) always wins. This is because of the sheer number of people playing, even though an individual has a low chance of success, overall it's almost certain that it will be won by somebody. Furthermore, most people will refrain from submitting a ticket with six sequential numbers because of the "rationalisation" that such a draw is too improbable. This is despite the fact that all draws are equally likely.
The idea of this can also be expressed by looking at car licence plates. Imagine seeing one with the configuration HJB-541. That's one out of a combination of over 17 million, so it seems like a remarkably improbable feat if you treat it in the same way as the statistically illiterate. But any combination is equally improbable, and you're certain to see one of the combinations if you look for it. It would only become remarkable if you predicted the configuration in advance.
Same birthday
Consider a party attended by thirty people: what are the chances that two of them have the same birthday? One in twelve, or, roughly 8% (30/365)? No, the odds are significantly better than that. In fact, there is a 70.6% probability. This is known as the "birthday problem". The apparently miraculous breaking of odds is attributed to the fact that the question is "what is the chance that any two people have the same birthday?", whereas most people following common sense tend to translate the question as "what is the chance that someone will have the same birthday as me?". Regardless, the answer is very non-intuitive and is a good display of how people don't do well at guessing probabilities.
At least one
Another example in statistics is "at least one". Imagine 6 cards are laid out face-down and the only certainty is that 2 cards are aces and 4 cards are not aces. What many people will assume from intuition is the chance of choosing at least one ace when flipping two cards over is 2 of 6 (~33%). The actual chance of picking up at least one ace is much better than that.
The probability of at least one is equal to 1 minus the probability of none. In this case the probability of not drawing any aces can be determined with the formula P(A)*P(A|B), which is read as "the probability of A multiplied by the probability of B assuming that event A already occurred" (since there is no replacement of the first card). So, P(A)*P(A|B) would be (4/6ths)*(3/5ths), which equals 12/30ths. 1 minus 12/30ths is 18/30ths, or 60%.
So, the probability of drawing at least one ace is 60% because the chance of drawing no aces is only 40%.
The amazing coin predictor
Imagine someone who could predict the random toss of a coin ten times in a row. Surely that person must be psychic or incredibly lucky?
No, they are just happen to be the random 1 in 1000 for whom the statistics are favorable.
If we have 1000 people, or for the sake of exact numbers 1024 people, who are asked to predict the toss of a coin. Assuming half will always call tails and the other half will call heads then 512 will get it right once. On a second toss half (256) of the first group will again guess right. Subsequent tosses will halve the number of people who get it right until after 10 tosses one person will have been right 10 times in a row. On such statistical streaks are the careers of stock-pickers and other charlatans built.
This effect was exploited in Derren Brown's special The System, where he presented a system for winning bets placed on multiple race horses. He began with several thousand volunteers and then subsequently only followed the winners; the final product that was televised only featured one individual, making his "system" seem miraculous. To demonstrate the system, he also performed the coin tossing trick, taking around 9 hours to film all of his attempts until he did come up with a successful combination.
Paul the Octopus
A similar thing happened in the 2010 South Africa World Cup, when Paul the Octopus was thought to have predicted the outcome of eight matches. A great part of the real explanation is very simple: there was a 1 in 256 probability that Paul could predict the outcome of eight games, and Paul just casually happened to be that one in 256. (Magical thinking, of course, processed this fact as Paul being a psychic octopus).
Shuffling a deck of cards
When you shuffle a deck of cards, they will end up in one of 52! [1] possible configurations. In a perfectly random shuffle of a perfect deck of cards, no one configuration is any more or less likely than any other. However, the odds are 100% that they will end up in some order, regardless of the odds against the particular order they end up in.
See also
- Borel's Law - a rough statistical rule of thumb often abused by creationists in argument
- Magical thinking
- Confirmation bias
- Gambler's fallacy
- Conservapedia:Probability
- Littlewood's law
Footnotes
- ↑ 52! is about 8.06581752×10^67. That's a very big number. And why we don't use wp:bogosort