The binomial distribution is a probability distribution by which the probability of the occurrence of events is calculated, provided that they occur under two modalities: success or failure.
These designations (success or failure) are completely arbitrary, as they do not necessarily mean good or bad things. During this article we will indicate the mathematical form of the binomial distribution and then the meaning of each term will be explained in detail.
Figure 1. The roll of a die is a phenomenon that can be modeled using the binomial distribution. Source: Pixabay.
Equation
The equation is as follows:
With x = 0, 1, 2, 3….n, where:
- P (x) is the probability of having exactly x successes between n attempts or trials.
- x is the variable that describes the phenomenon of interest, corresponding to the number of successes.
- n the number of attempts
- p is the probability of success in 1 attempt
- q is the probability of failure in 1 attempt, therefore q = 1 - p
The exclamation mark "!" is used for factorial notation, so:
0! = 1
one! = 1
two! = 2.1 = 2
3! = 3.2.1 = 6
4! = 4.3.2.1 = 24
5! = 5.4.3.2.1 = 120
And so on.
Concept
The binomial distribution is very appropriate for describing situations in which an event occurs or does not occur. If it occurs it is a success and if not, then it is a failure. Furthermore, the probability of success must always remain constant.
There are phenomena that fit these conditions, for example the toss of a coin. In this case, we can say that "success" is getting a face. The probability is ½ and does not change, no matter how many times the coin is tossed.
The roll of an honest die is another good example, as well as categorizing a certain production into good pieces and defective pieces and obtaining a red instead of a black when spinning a wheel.
characteristics
We can summarize the characteristics of the binomial distribution as follows:
- Any event or observation is extracted from an infinite population without replacement or from a finite population with replacement.
- Only two options are considered, mutually exclusive: success or failure, as explained at the beginning.
- The probability of success must be constant in any observation that is made.
- The result of any event is independent of any other event.
- The mean of the binomial distribution is np
- The standard deviation is:
Application example
Let's take a simple event, which may be getting 2 heads 5 by rolling an honest die 3 times. What is the probability that in 3 tosses 2 heads of 5 will be obtained?
There are several ways to achieve this, for example:
- The first two launches are 5 and the last is not.
- The first and the last are 5 but not the middle one.
- The last two throws are 5 and the first does not.
Let's take the first sequence described as an example and calculate its probability of occurrence. The probability of getting a 5 heads on the first roll is 1/6, and also on the second, as they are independent events.
The probability of getting another head other than 5 on the last roll is 1 - 1/6 = 5/6. Therefore, the probability that this sequence comes out is the product of the probabilities:
(1/6). (1/6). (5/6) = 5/216 = 0.023
What about the other two sequences? They have the same probability: 0.023.
And since we have a total of 3 successful sequences, the total probability will be:
Example 2
One university claims that 80% of students on the college basketball team graduate. An investigation examines the academic record of 20 students belonging to said basketball team who enrolled in the university some time ago.
Of these 20 students, 11 finished their studies and 9 dropped out.
Figure 2. Almost all students who play for the college team graduate. Source: Pixabay.
If the university's statement is true, the number of students who play basketball and graduate, out of 20, should have a binomial distribution with n = 20 and p = 0.8. What is the probability that exactly 11 of the 20 players will graduate?
Solution
In the binomial distribution:
Example 3
The researchers conducted a study to determine whether there were significant differences in graduation rates between medical students admitted through special programs and medical students admitted through regular admission criteria.
The graduation rate was found to be 94% for student physicians admitted through special programs (based on data from the Journal of the American Medical Association).
If 10 of the special programs students are randomly selected, find the probability that at least 9 of them graduated.
b) Would it be unusual to randomly select 10 students from special programs and find that only 7 of them have graduated?
Solution
The probability that a student admitted through a special program will graduate is 94/100 = 0.94. We choose n = 10 students from the special programs and we want to find out the probability that at least 9 of them will graduate.
The following values are then substituted in the binomial distribution:
b)
References
- Berenson, M. 1985. Statistics for Management and Economics. Interamericana SA
- MathWorks. Binomial distribution. Recovered from: es.mathworks.com
- Mendenhall, W. 1981. Statistics for Management and Economics. 3rd. edition. Grupo Editorial Iberoamérica.
- Moore, D. 2005. Applied Basic Statistics. 2nd. Edition.
- Triola, M. 2012. Elementary Statistics. 11th. Ed. Pearson Education.
- Wikipedia. Binomial distribution. Recovered from: es.wikipedia.org