PROBABILITY is a branch of mathematics that deals with the likelihood or chance of different outcomes occurring in uncertain situations. It quantifies how likely an event is to happen and is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
Probabilities are important to understand because they give us predictive capabilities: probabilities are the closest thing we have to being able to predict the future. Of course, this is not foolproof; there is always a chance that we are wrong, which is why we couch results/findings in terms of a 95% confidence interval and margin of error.
Basic Law of Probability
“The BASIC LAW OF PROBABILITY . . . states the following: Given that all possible outcomes of a given event are equally likely, the probability of any specific outcome is equal to the ratio of the number of ways that that outcome could be achieved to the total number of ways that all possible outcomes can be achieved” (Meier, Brudney, and Bohte, 2011, p. 113). This means that we can predict the outcome of a specific event as long as the likelihood of a given event is known. For example:
- the probability of getting heads with one coin flip is 1/2
- the probability of getting three with one roll of a six-sided dice is 1/6
- the probability of drawing the ace of spades from a deck of cards is 1/52
- the probability of drawing an ace from a deck of cards is 1/13 (although there are 52 cards in a deck, there are four aces; to calculate the probability of drawing an ace from a deck of cards, you would divide 52 by four)
- the probability of drawing a heart from a deck of 52 cards is 1/4 (this is because there are four suits in each deck of cards)
All of these examples involve probabilities of the occurrence of single events. As you can see, the process of calculating these probabilities is pretty straightforward: you divide the number of times a specific outcome can occur by the total number of possible outcomes.
Probability P(A) of a Single Event A
P (A) = Number of favorable outcomes / Total number of possible outcomes
This gets a little more complicated as we factor in other events; the manner in which we would calculate the probability of an event occurring differs depending on whether we are looking at mutually exclusive events (i.e., events that cannot occur at the same time), non-mutually exclusive events (i.e., events that can occur at the same time), independent events (i.e., events in which the occurrence of one does not affect the occurrence of the other), an event occurring given the occurrence of a different event (i.e., the conditional probability), etc. Nevertheless, while different equations are used to calculate probabilities in these situations, the basic law of probability still exists: the probability of an event occurring falls between 0 (“never occurs”) and 1 (“always occurs”), and calculating this probability is based on possible outcomes.
A Priori and Posterior Probabilities
A PRIORI PROBABILITIES are initial probabilities of an event based on existing knowledge, theory, or general reasoning about the event. Everything we have discussed thus far are a priori probabilities, because we know the possible outcomes of a coin flip, die roll, or card draw. By contrast, POSTERIOR PROBABILITIES are probabilities of an event after new evidence or information is taken into account. A classic example of posterior probability is the Monty Hall problem. Posterior probabilities are often calculated using BAYES’ THEOREM, which combines the prior probability with the likelihood of new evidence or information. Frequency distributions provide the empirical data needed to estimate the probabilities used in Bayes’ theorem.