Estimating the Odds: Probability 101

Consider the following statements.

  • I’ll probably have dinner at my favorite restaurant next week
  • My chances of a raise next month are pretty good, maybe 40-60
  • The probability of drawing the queen of hearts from a random card deck is 1 in 52

We all have intuitive ideas about probability or chance since we make life decisions on a regular basis, attempting to maximize our happiness or success based on estimated outcomes. But in real life, including marketing applications of probability, subtle issues often surface.

We often guess that the probability of some future event is based by past experience. Maybe the bank teller Fred has only missed 3 days over the past 1000 work days so we think that the probability that he will show up tomorrow is 0.997. But in point of fact, Fred may have died last night. Had we known this critical fact our probability estimate of his coming to work would have been zero.

This general idea of probability and information is formalized in genuine scientific applications by making sharp distinctions between the abstract probability P of an event and the estimated probability, using the symbol P-hat (P with a little hat sign on top, p?). In many cases we can establish error bounds on the actual probability P based on our P-hat estimate, plus the all other known information. Thus, P-hat always depends on our knowledge and ignorance.

To illustrate the difference between estimated and actual probability, consider the example of a simple coin toss that might be biased in some unknown manner. What is the probability of obtaining a head? Let’s do an experiment—suppose we toss the coin 100 times and obtain 61 heads. One plausible estimate of the probability of heads is 61/100 or 0.61. But suppose we then look more closely at the coin, and the two sides appear to be symmetric, implying that a better estimate of heads may be 50/100 or 0.5. We also do a calculation (based on the famous bell curve) and find that if the coin is indeed not biased (meaning probability of heads equal to 0.5), the probability of us getting heads 61 or more times in 100 tosses is 0.018. Thus, our analysis implies that there is a 0.018 chance (or 2% rounded) which is 1 in 50 that the coin is not biased and conversely, a chance of 49 in 50 that the coin is biased in some way towards heads.

The main lesson is this—even with all this information, the actual probability of heads for this coin is still unknown. We may improve our estimate by more coin tosses and by means of a closer study of the coin’s physical characteristics and the manner in which it is tossed. But we can never know the actual probability of heads. We only know our estimates.

How Do These Ideas Apply to Marketing Research?

Suppose I am thinking offering a new product, changing the price of an existing service, or opening a new retail store. I woudl like to know the probability that any of these changes will generate $Y or more in income over the next three years. Or perhaps I would like to know the probability of different populations purchasing my product or service. But, as the simple coin toss example implies, this probability is fundamentally unknowable. However, by employing appropriate combinations of qualitative and quantitative research, we can find the best possible estimate of this critical probability. The specifics of the research process will, of course, differ according to the actual business environment. As in most research, the devil is in the details.

Paul L. Nunez, Ph.D. is Vice President of Neuroscience Applications at Q2 Insights, Inc., a marketing research consulting firm with offices in San Diego and New Orleans.  Paul is also Emeritus Professor, Tulane University and the author of four books on the neuroscience of EEG published by Oxford University Press, an upcoming book on Consciousness that is being published by Prometheus Books, and over 100 peer reviewed academic papers on the same topic.  He can be reached at (760) 230-2950 or at

This entry was posted in Data Analysis and tagged on September 7, 2016 by brett_adm