Assume that a procedure yields a binomial distribution with a trial repeated n times. Using the binomial probability formula, what is the probability of x successes given the probability p of success on a single trial? Round your answer to three decimal places.

Assuming this complete question :"Assume that a procedure yields a binomial distribution with a trial repeated n times. Using the binomial probability formula, what is the probability of x successes given the probability p of success on a single trial? Round your answer to three decimal places.

n=30, x= 5, p=1/5"

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Solution to the problem

Let X the random variable of interest, on this case we know that:

[tex]X \sim Binom(n=30, p=0.2)[/tex]

The probability mass function for the Binomial distribution is given as:

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]

Where (nCx) means combinatory and it's given by this formula:

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]

And for this case if we find the probability for x=5 we got:

[tex]P(X=5)=(30C5)(0.2)^5 (1-0.2)^{30-5}=0.172[/tex]