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."
This is because once you know a face card has been drawn, 4 out of the 12 face cards are kings, giving a probability of 4/12, or 1/3.
Conditional probabilities are often the result of narrowing the population of interest. Consider the following statements (using made-up figures):
- "% of the population are atheists."
- "% of the population are convicted criminals."
- "% of convicted criminals are atheists."
Given this information, does it look like atheism is associated with criminal conviction? Notice that the third percentage is calculated out of a different total: only atheists instead of the whole population. That means the percents are not directly comparable.
The questions we really need to ask to get an accurate impression of what's going on are the following:
- "What percent of atheists are convicted criminals?"
- "What percent of theists are convicted criminals?"
Here's a table that matches the above information. All percents are now out of the total population.
Notice that statements #1 and #2 above are obviously true based on the table (look in the "Total" column and "Total" row, respectively). To verify statement #3:
- Fraction of convicted criminals that are atheists = %
Now, to answer the two questions above:
- Fraction of atheists who are convicted criminals = %
- Fraction of theists who are convicted criminals = %
To be finished later...