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Binomial distributions 101

XXX

Fair coin

				probability
H	H	H	H
H	H	H	T
H	H	T	H
H	H	T	T
H	T	H	H
H	T	H	T
H	T	T	H
H	T	T	T
T	H	H	H
T	H	H	T
T	H	T	H
T	H	T	T
T	T	H	H
T	T	H	T
T	T	T	H
T	T	T	T

XXX

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.

First, we let "n" denote the number of observations or the number of times the process is repeated, and "x" denotes the number of "successes" or events of interest occurring during "n" observations. The probability of "success" or occurrence of the outcome of interest is indicated by "p".

The binomial equation also uses factorials. In mathematics, the factorial of a non-negative integer k is denoted by k!, which is the product of all positive integers less than or equal to k. For example,

The binomial distribution model

P(X=x) = \binom{n}{x} \cdot p^x \cdot \left(1-p \right)^{n-x}

Which can be rewritten as

P(X=x) = \frac{n!}{x!(n-x)!} \cdot p^x \cdot \left(1-p \right)^{n-x}

A few things to mention here

p: XXX
n: number of trials
x: the result we are analyzing

Testing a new medicine

A pharmaceutical company is testing a new medicine against X. The probability that the medicine is effective is 76.4%. In a clinical trial, 7 patients undergo the treatment. What is the probability that 4 of them get cured?

Analysis

First of all, we must define our discrete random variable. If we define it as the number of patients that get cured, the probability of success (or the probability that happens) is $p = 0.764$ . Also, given we want to calculate the chances that 4 of them get cured (which produces the sample space seen below), $x = 4$ . We'll use the formula to verify the result we calculate by a brute force method

It is easy to see that doing it manually will require a lot more time that using our formula.

How would that probability change if