The binomial distribution is a probability distribution by which the probability of 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.
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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
- what is the probability of failure in 1 attempt, therefore q = 1 - p
The exclamation mark "!" is used for factorial notation, so:
0! = 1
1! = 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.
The binomial distribution is very appropriate to describe 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. In addition, 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..
We can summarize the characteristics of the binomial distribution as follows:
- Any event or observation is drawn from an infinite population without replacement or from a finite population with replacement.
- Only two mutually exclusive options are considered: success or failure, as explained at the beginning.
- The probability of success must be constant in any observation made.
- The result of any event is independent of any other event.
- The mean of the binomial distribution is n.p
- The standard deviation is:
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 rolls are 5 and the last is not.
- The first and 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, since 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:
P (2 heads 5 in 3 tosses) = Number of possible sequences x probability of a particular sequence = 3 x 0.023 = 0.069.
Now let's try the binomial, in which it is done:
x = 2 (getting 2 heads of 5 in 3 tosses is success)
n = 3
p = 1/6
q = 5/6
There are several ways to solve the binomial distribution exercises. As we have seen, the simplest can be solved by counting how many successful sequences there are and then multiplying by the respective probabilities.
However, when there are many options, the numbers get larger and it is preferable to use the formula.
And if the numbers are even higher, there are tables of the binomial distribution. However, they are now obsolete in favor of the many kinds of calculators that facilitate calculation..
A couple has children with a probability of 0.25 of having type O blood. The couple has a total of 5 children. Answer: a) Does this situation fit a binomial distribution? B) What is the probability that exactly 2 of them are of type O?
a) The binomial distribution is adjusted, since it meets the conditions established in previous sections. There are two options: having type O blood is "success", while not having it is "failure", and all observations are independent..
b) We have the binomial distribution:
x = 2 (obtain 2 children with type O blood)
n = 5
p = 0.25
q = 0.75
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.
If the university's claim is true, the number of students who play basketball and graduate, out of 20, should have a binomial distribution with n = 20 Y p = 0.8. What is the probability that exactly 11 of the 20 players graduate??
In the binomial distribution:
x = 11
n = 20
p = 0.8
q = 0.2
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 medical students 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??
The probability that a student admitted through a special program will graduate is 94/100 = 0.94. Are chosen n = 10 students from special programs and you want to find out the probability that at least 9 of them graduate.
The following values are then substituted in the binomial distribution:
x = 9
n = 10
p = 0.94
b)
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