Probability: Difference between revisions
imported>Greg Woodhouse (Bayesian probability - brief explanation) |
imported>Greg Woodhouse (Objective probability - rewrite) |
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How likely is it that it will rain today? If I'm inside a room with no windows and cannot look outside and see whether there are any clouds in the sky or wshether the wind is blowing, then I do not have this information available, and cannot use it to inform my estimate of how likely it is that it going to rain. But of the last 100 cloudy days I've experienced, I've noticed that it rained on 20 of them, and that of days without a cloud in the sky it has only rained on five (because a storm blew in later in the day), I will conclude that rain is more likely on cloudy days. In fact, this can be made precise using a formula known as [[Bayes' theorem]] which expresses the probability of rain ''given'' that it is cloudy in terms of the probability that it will rain on a cloudy day (something I can estimate by direct observation), the probability that it will rain on a given day, and the probability that it will be cloudy on a given day (both of which I can estimate by direct observation). | How likely is it that it will rain today? If I'm inside a room with no windows and cannot look outside and see whether there are any clouds in the sky or wshether the wind is blowing, then I do not have this information available, and cannot use it to inform my estimate of how likely it is that it going to rain. But of the last 100 cloudy days I've experienced, I've noticed that it rained on 20 of them, and that of days without a cloud in the sky it has only rained on five (because a storm blew in later in the day), I will conclude that rain is more likely on cloudy days. In fact, this can be made precise using a formula known as [[Bayes' theorem]] which expresses the probability of rain ''given'' that it is cloudy in terms of the probability that it will rain on a cloudy day (something I can estimate by direct observation), the probability that it will rain on a given day, and the probability that it will be cloudy on a given day (both of which I can estimate by direct observation). | ||
== | == Frequentist probability == | ||
In this approach one views probabilities as | In this approach one views probabilities as the proportion of identical (or as nearly identical as we can manage) experiments will have a given outcome. | ||
=== Example of the frequentist viewpoint === | |||
The classic example here is flipping a coin. If out of 1000 coin flips, 501 are "heads" and 499 are "tails", a frequentist will say that (based on this experiment) the probability of heads is .501. Now, if the outcome is truly random, then if we flip the coin 10,000 times (or 100,000 times), the proportion of heads will come even closer to .5. The .501 we derived by flipping our coin 1000 times is only an estimate of the true probability. The difficulty is that we can only carry out experiments a finite number of times, so the frequentist approach doesn't tell us exactly what the probability should be, either. | |||
== More technical information == | == More technical information == | ||
Revision as of 21:05, 14 May 2007
Probability is a number representing an estimate of how likely an event is, ranging from 1.0 representing certainty down to 0. for impossibility.
Probability is the topic of probability theory, a branch of mathematics concerned with analysis of random phenomena. Like algebra, geometry and other parts of mathematics, probability theory has its origins in the natural world. Humans routinely deal with incomplete and/or uncertain information in daily life: in decisions such as crossing the road ("will this approaching car respect the red light?"), eating food ("am I certain this food is not contaminated?"), and so on. Probability theory is a mathematical tool intended to formalize this ubiquitous mental process. The probability concept is a part of this theory, and is intended to formalize uncertainty.
There are two basic ways to think about the probability concept:
- Bayesian probability.
- Frequentist ("objective") probability.
The different approaches are largely pedagogical, as some people find one approach or the other much easier.
Bayesian probability
In this approach, probability is taken as a measure of how reasonable a belief is in light of experience or observations. It is based on a rigorous relationship between what are called conditional probabilities and ordinary (non-conditional) probability. It is thus, not simply an intuitive or educated "guess", but something much more specific and precise.
Example of the Bayesian viewpoint
How likely is it that it will rain today? If I'm inside a room with no windows and cannot look outside and see whether there are any clouds in the sky or wshether the wind is blowing, then I do not have this information available, and cannot use it to inform my estimate of how likely it is that it going to rain. But of the last 100 cloudy days I've experienced, I've noticed that it rained on 20 of them, and that of days without a cloud in the sky it has only rained on five (because a storm blew in later in the day), I will conclude that rain is more likely on cloudy days. In fact, this can be made precise using a formula known as Bayes' theorem which expresses the probability of rain given that it is cloudy in terms of the probability that it will rain on a cloudy day (something I can estimate by direct observation), the probability that it will rain on a given day, and the probability that it will be cloudy on a given day (both of which I can estimate by direct observation).
Frequentist probability
In this approach one views probabilities as the proportion of identical (or as nearly identical as we can manage) experiments will have a given outcome.
Example of the frequentist viewpoint
The classic example here is flipping a coin. If out of 1000 coin flips, 501 are "heads" and 499 are "tails", a frequentist will say that (based on this experiment) the probability of heads is .501. Now, if the outcome is truly random, then if we flip the coin 10,000 times (or 100,000 times), the proportion of heads will come even closer to .5. The .501 we derived by flipping our coin 1000 times is only an estimate of the true probability. The difficulty is that we can only carry out experiments a finite number of times, so the frequentist approach doesn't tell us exactly what the probability should be, either.
More technical information
- Bayes theorem
- principle of maximum entropy
- Probability distributions
- Kolmogorov's axioms
- probability theory
Links
External links
- Intros