A binomial process involves a lot of assumptions. For example, all the trials ar

Question

A binomial process involves a lot of assumptions. For example, all the trials are supposed to be independent, have only two outcomes, and repeated under identical conditions with the same probability of success and failure each time. Yet we can't always be certain that the probability of success won't change from one trial to the next. In the real world, there is almost nothing we can be absolutely sure about, so the theoretical assumptions of the binomial probability distribution will not be completely satisfied. Does that mean we can never use the binomial probability distribution? The answer seems to be that we can indeed use the binomial distribution even if not all the assumptions are exactly met. List three applications of the binomial distribution, for which you think there is adequate reason to apply the binomial distribution even if some of the assumptions are not exactly met.

Explanation / Answer

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The binomial distribution model is an important probability model that is used when there are two possible outcomes (hence "binomial"). In a situation in which there were more than two distinct outcomes, a multinomial probability model might be appropriate, but here we focus on the situation in which the outcome is dichotomous.

For example, adults with allergies might report relief with medication or not, children with a bacterial infection might respond to antibiotic therapy or not, adults who suffer a myocardial infarction might survive the heart attack or not, a medical device such as a coronary stent might be successfully implanted or not. These are just a few examples of applications or processes in which the outcome of interest has two possible values (i.e., it is dichotomous). The two outcomes are often labeled "success" and "failure" with success indicating the presence of the outcome of interest. Note, however, that for many medical and public health questions the outcome or event of interest is the occurrence of disease, which is obviously not really a success. Nevertheless, this terminology is typically used when discussing the binomial distribution model. As a result, whenever using the binomial distribution, we must clearly specify which outcome is the "success" and which is the "failure".

The binomial distribution model allows us to compute the probability of observing a specified number of "successes" when the process is repeated a specific number of times (e.g., in a set of patients) and the outcome for a given patient is either a success or a failure. We must first introduce some notation which is necessary for the binomial distribution model.

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