Probability is a concept that is generally easy to understand intuitively, but can be difficult to define rigorously. When it is defined and studied carefully, it can lead to counter-intuitive results. Even elementary ideas of probability can be widely misunderstood, making it a good way to dishonestly bolster a weak argument.
When properly used, probability and its more "applied" cousin, statistics, can be powerful tools for discerning empirical truth. Indeed, essentially every field of modern science relies heavily on statistical analysis of data, which in turn relies on probability.
Definitions of probability
There are two main interpretations of probability, each of which leads to a different definition of probability:
- Relative frequency
- The long-run relative frequency of occurrence of a random event (the fraction of the time it happens in a long run of repeated "trials") can be defined as the probability of the event.
- Example: "A 'fair' coin has a 50% probability of coming up 'heads'."
- More information: Wikipedia:Frequency probability
- Degree of belief (a.k.a., personal probability)
- The degree to which one believes a statement to be true can be defined as the probability of that statement.
Although the degree-of-belief "definition" might seem quite weak, it can be made rigorous by carefully considering, for example, how much one would be willing to bet in a game where one would gain a certain amount of money if the statement turns out to be true. It can be shown that any internally consistent method of choosing one's wager must obey the laws of probability.
One of the most widely misunderstood concepts in probability has to do with conditional probability, the probability of one thing happening (or being true) given that something else definitely happens (or is true).
- Example: "The probability that a 'king' has been drawn from a well-shuffled deck of playing cards given that you know it is a face card is one-third."