# Probability

… a core concept used in Quantitative Methods and Atlas104

### Concept description

Wikipedia (reference below) defines probability as the measure of the likelihood that an event will occur.

OnlineStatBook (link on right) notes that “inferential statistics is built on the foundation of probability theory [yet] the very idea of probability has been plagued by controversy from the beginning of the subject to the present day.”

OnlineStatBook notes that:

“One conception of probability is drawn from the idea of symmetrical outcomes. For example, the two possible outcomes of tossing a fair coin seem not to be distinguishable in any way that affects which side will land up or down. Therefore the probability of heads is taken to be 1/2, as is the probability of tails. In general, if there are N symmetrical outcomes, the probability of any given one of them occurring is taken to be 1/N. Thus, if a six-sided die is rolled, the probability of any one of the six sides coming up is 1/6.

“Probabilities can also be thought of in terms of relative frequencies. If we tossed a coin millions of times, we would expect the proportion of tosses that came up heads to be pretty close to 1/2. As the number of tosses increases, the proportion of heads approaches 1/2. Therefore, we can say that the probability of a head is 1/2. …

“For some purposes, probability is best thought of as subjective. Questions such as “What is the probability that Ms. Garcia will defeat Mr. Smith in an upcoming congressional election?” do not conveniently fit into either the symmetry or frequency approaches to probability. Rather, assigning probability 0.7 (say) to this event seems to reflect the speaker’s personal opinion – perhaps his willingness to bet according to certain odds. Such an approach to probability, however, seems to lose the objective content of the idea of chance; probability becomes mere opinion.

“Two people might attach different probabilities to the election outcome, yet there would be no criterion for calling one “right” and the other “wrong.” We cannot call one of the two people right simply because she assigned higher probability to the outcome that actually transpires. After all, you would be right to attribute probability 1/6 to throwing a six with a fair die, and your friend who attributes 2/3 to this event would be wrong. And you are still right (and your friend is still wrong) even if the die ends up showing a six! The lack of objective criteria for adjudicating claims about probabilities in the subjective perspective is an unattractive feature of it for many scholars.”

###### Atlas topic, subject, and course

Probability Theory (core topic) in Quantitative Methods and Atlas104 Quantitative Methods.

###### Sources

Wikipedia, Probability, at https://en.wikipedia.org/wiki/Probability, accessed 10 June 2017.

Dan Osherson, OnlineStatBook, at http://onlinestatbook.com/2/probability/probability_intro.html, accessed 10 June 2017.