In communication science, various scenarios can be modeled using the binomial distribution. For example, researchers might investigate whether a particular communication strategy increases the likelihood of a desired audience response, such as positive feedback, engagement, or participation. Other applications include examining audience recall, testing message effectiveness, or analyzing media consumption behavior where there are two possible outcomes.
Consider the following cases.
Case 1
A researcher in communication science is studying the effectiveness of a new social media campaign designed to minimize misinformation about a public health issue. The campaign targets a specific population, and the researcher wants to know the probability that a certain number of individuals in that population correctly identify accurate information after seeing the campaign. Previous similar research concluded that 60% of individuals in the population could distinguish accurate information from inaccurate information after seeing the campaign. If 10 individuals are randomly selected from the population, what is the probability that exactly 7 individuals can distinguish accurate from inaccurate information?
Case 2
A political campaign is testing a new slogan for its candidate. They hold focus groups with 12 participants and observe whether each participant finds the slogan memorable. Based on previous similar campaigns, they estimated that the probability that someone would find the slogan memorable was 40%. What is the probability that exactly 5 out of 12 participants in the focus group would find the new slogan memorable?”
The questions raised in both cases can be answered by understanding a very important probability distribution in statistics, the binomial distribution. To begin our discussion of this distribution, let’s ask: “What do the two cases have in common?” In the first case, the possible outcomes are “can distinguish” and “cannot distinguish” two possibilities. In the second case, the possible outcomes are “impressive” and “unimpressive,” also two possibilities. The word binomial comes from the Latin binomius, a combination of bi, meaning two, and nomen, meaning name.
A GLIMPSE OF THE BINOMIAL DISTRIBUTION
In short, the binomial distribution calculates the probability of a specific number of successes in a given number of observations. In Case 1, the question is how likely it is that 7 people (out of 10) “can distinguish.” In this example, “can distinguish” is considered a success, while “cannot distinguish” is considered a failure. So if exactly 7 people “can distinguish” accurate information from inaccurate information, then there are 7 successes. The number of observations in this case is 10. In Case 2, the question is how likely it is that exactly 5 people (out of 12 participants) will find the new slogan “impressive.” In this example, “impressive” is considered a success, while “not impressive” is considered a failure. So, if exactly 5 people find the new slogan “impressive,” then there are 5 successes. The number of observations in this case is 12.
Let us discuss Case 2 further. Here, the probability that a discussion participant finds the slogan impressive is 40%. So, for each participant in the discussion, there is uncertainty whether the participant is impressed or not. There are 12 participants in the discussion. For each participant, we do not know whether the participant is impressed or not with the new slogan. It is possible that among all participants, only 2 people are impressed. It is also possible that 5 people are impressed. It is also possible that 9 people are impressed, and so on. Each of these possibilities has a probability of happening. But how big is the probability? This is where the binomial distribution may help.
Let us answer the question in Case 2 with the help of Microsoft Excel. In this question, what is the probability that there are 5 people who are impressed? To find the answer, type the following expression into an Excel spreadsheet cell: =BINOM.DIST(5;12;0.4;0). (See Figure 1.)
Figure 1
After the Enter key is pressed, the value 0.227 will appear (Figure 2). So, in Case 2, the probability that exactly 5 people are impressed is 22.7%.
Figure 2
In the expression =BINOM.DIST(5;12;0.4;0), the value 5 is the number of successes, 12 is the number of observations, 0.4 is the probability of success (in this case: “impressed”), and 0 is the code that what is calculated is the probability of exactly 5 successes, not ≤ 5 successes.
To answer the question in Case 1, type the expression =BINOM.DIST(7;10;0.6;0). After the Enter key is pressed, the value 0.215 will appear. So, the probability that exactly 7 people can distinguish accurate information from inaccurate information is 21.5%. In the expression =BINOM.DIST(7;10;0.6;0), the value 7 is the number of successes, 10 is the number of observations, 0.6 is the probability of success (in this case: “can distinguish”), and 0 is a code that the probability of exactly 7 successes is being calculated, not ≤ 7 successes.
Case 3
A researcher tests whether viewers of a TV commercial can remember the message of the advertisement. A group of 15 participants are exposed to the advertisement, and then each participant is asked whether they remember the message. From previous research, it is known that the probability of a participant remembering the message of such an advertisement is 33%. What is the probability that among the 15 participants: a) exactly 4 people remember the message, b) no more than 4 people remember the message.
Here, for each viewer, there are exactly two outcomes: “the participant remembers the message of the advertisement” or “the participant does not remember the message of the advertisement”. Thus, in this case, the binomial property applies. If a viewer remembers it, we assume that a success has occurred. On the other hand, if a viewer does not remember it, then we assume that there is a failure. We do not know how many of the 15 viewers remember the message of the advertisement. We do not know how many successes occurred, out of the 15 observations. It is possible that exactly 4 viewers remember the message of the advertisement. What is the probability that this will happen? This is what is answered by the binomial distribution. In a Microsoft Excel cell, enter this expression: =BINOM.DIST(4;15;0.33;0). After the Enter key is pressed, the value 0.198 will appear. So, the probability that exactly 4 people can remember the message is 19.8%. In the expression =BINOM.DIST(4;15;0.33;0), the value 4 is the number of successes, 15 is the number of observations, 0.33 is the probability of success (in this case: “viewers can remember the message”), and 0 is the code that what is being calculated is the probability of exactly 4 successes, not ≤ 4 successes. Therefore, the answer to part a) is 0.198 or 19.8%. To deal with part b), type the following expression: =BINOM.DIST(4;15;0.33;1). After the Enter key is pressed, the value 0.415 will appear. So, the probability that no more than 4 people can remember the message is 41.5%. In the expression =BINOM.DIST(4;15;0.33;1), the value 1 is a code that what is calculated is the probability of ≤ 4 successes.
The previous answer, 0.415, can also be obtained by the following thinking process. The occurrence of no more than 4 successes is identical to a compound event composed of 5 mutually exclusive single events. The five single events are: “4 successes occur”, “3 successes occur”, “2 successes occur”, “1 success occurs”, and “0 successes occur”. By using the appropriate expressions, the probability of each single event is obtained as follows.
Figure 3
Then, using the Excel function =SUM(), add up the five probability values, as follows.
Figure 4
After the Enter key is pressed, the value 0.415 will appear.
The Assumptions of The Binomial Distribution
Indeed, not in all cases we can apply the binomial distribution. There are several assumptions for this distribution to apply. More detailed material on this and the assumptions of its applications is available in the following link: (click here)
