The Scientific "Maybe" : A Probability-Possibility Dilemma

There is without doubt that most of us employ these concepts in our day to day activities, sometimes even interchanging one for the other, while some will take a bolder step and assume them to be the same. What is probability and what is possibility? Are they the same? Are they related in any way? How do these concepts affect you and I as individuals? These conundrums will be clarified in this article as we take a deeper look into the possibility-probability relationship.

Probability or Possibility

James woke up one morning and suddenly began asking himself, "what if I die tomorrow?"
Granted, everyone would die someday, but how sure is he that that it would be tommorow. Notice that this is however different from James asking himself the question, "what if I don't wake up from my sleep?" or "what is the likelihood of me eating pancakes this morning?" which is quite different from the question, "what is the likelihood of me eating pancakes this morning out of a hamper of cakes, doughnuts and pancakes?"
How then do these questions differ when they seem so much alike? We will get to that in a minute.

Probability

Probability as defined by the Merriam Webster dictionary is "A measure of how often an a measure of how often a particularevent will happen if something (such as tossing a coin) is done repeatedly." It also defines it implicitly as "the chance that a given event will occur."
Mathematically, Probability is defined as the "the ratio of the number of outcomes in an exhaustive set of equally likely outcomes that produce a given event to the total number of possible outcomes.
Put simply, probability is the quality or condition of being probable
It is this definition that we find the most intrinsic and distinguishing property of the probability concept : it is quantifiable. Mathematically, the quantification can be seen as a range bar with 0 at one end and 1 at the other end where 0 implies an impossibility and 1 represents certainty. Therefore, in a situation where an action is repetitive, a probability closer to 0 on the range bar indicates an that an event is less likely to occur if the action is repeated while a probability closer to 1 indicates a likelihood for the event to occur. If James gets ready for work and would have to choose between wearing his black shoes and brown shoes, if we have an idea of what shoe James wore the day before and the day before that, we could from the application of probability infer what shoe James might put on today, i.e, if James had worn the brown shoe on Monday and the black shoe on Tuesday and the brown shoe on Wednesday, then it is highly probably that he would wear the black shoe on Thursday but not certain. Therefore in this case, the probability of James wearing a black shoe would be slightly greater that 1/2, which indicates a likelihood. However, if James had only one shoe and the shoe was coloured brown, then the probabilty of James wearing a Brown shoe to work would be highly certain and would be quantified as 1 while the probability of James wearing a black shoe is highly improbable because James does not have a black shoe, therefore the probability would be quantified as 0.

Negative probability

Negative probability is not a usual occurrence but is usually said to occur in quasiprobabilistic distributions. These distributions are usually found in the application of quantum mechanics and and time-frquency analysis and also in finance where the concepts of negative money arises. In his book, Interpretations of Negative Probabilities, Mark Burgins gave an example of what might be inferred as negative probability :

Let us consider the situation when an attentive person A with the high knowledge of English writes some text T. We may ask what the probability is for the word “texxt” or “wrod” to appear in his text T. Conventional probability theory gives 0 as the answer. However, we all know that there are usually misprints. So, due to such a misprint this word may appear but then it would be corrected. In terms of extended probability, a negative value (say, −0.1) of the probability for the word “texxt” to appear in his text T means that this word may appear due to a misprint but then it’ll be corrected and will not be present in the text T.

Negative probabilities are also seen in both finance (also known as risk neutral probabilities) and in engineering as well where retrogression and risk induced Correlations are modelled when facility locations, customer allocation, and backup service plans are determined simultaneously. Thus negative probabilities are known as puesdo probabilities or conditional probabilities where later occurrences might be taken into consideration allowing the probablistic value to exceed unity in case of assumed conditional occurrences. Take for example, in the James-shoe dilemma, in the case of James having only a pair of brown shoes and no black shoe, the probability of him wearing a black shoe is 0 but can be extended to -1 under the condition that on his way to work, James might branch into one of the boutiques along the way and get himself a black shoe. This is a quasiprobability, and is in some way much closer to possibility except that it is in this case, quantifiable.

Possibility

Possibility can be defined as a chance that something might exist, happen, or be true. It in simple terms, "the state or fact of being possible." the concept of possibility puts event in prospective values but not quantifiable and not in relation with other events. In other words, possibility cannot be measured and is independent of other events.
In our earlier example, where James wakes up and ask himself, "what if I die tomorrow?"
However, this question has only a prospective value and cannot be quantified because, yes, it is possible that James might die tomorrow just as well as it is possible for James to die tomorrow from a thunderstorm or a stroke. But what if James was a healthy as a horse and has no history of stroke in his family line, then although it is possible that James does from stroke tommorow, it is highly improbable. Just as it is highly improbable from him to die from a thunderstorm if the weather tomorrow would be above 40 degrees although it is possible. Similar, the question of James eating pancake in the morning is a possibility but how probable we do not know simply because we have no relative event or any recurrence to measure by what degree of possibility is the event of James eating a pancake in the morning. However, in a hamper of cakes, doughnuts and pancakes, then we can infer that the probability of James eating pancakes in the morning is 1/3, just as it is probable for him to eat cake or doughnuts.

The concept of probabilities and possibilities, however complicated can be simplified thus:
probability is the measure of the degree of possibility of an occurrence of an event.

I hope I have cleared the quandary on the differences between the probable concept and the possible concept. Feel free to make your contributions and do remember to stay scientific, always.

Reference list

1. Mark Burgin, Burgin, Mark (2010). "Interpretations of Negative Probabilities".

2.Haug, E. G. (2004): Why so Negative to Negative Probabilities. Wilmott Magazine, Re-printed in the book (2007); Derivatives Models on Models, John Wiley & Sons, New York.

3.Cui, T.; Ouyang, Y.; Shen, Z-J. M. (2010). Reliable Facility Location Design Under the Risk of Disruptions. Operations Research. 58 (4): 998–1011.