Question

I need to use factorial the! When you use the binomial probability formula and I need to show step-by-step for for a through c


Given the number of trials and the probability of success, determine the probability indicated: (Hint use binomial distribution formula use factorials ! in showing your work)

n = 15, p = 0.4, find P(4 successes)
n = 12, p = 0.2, find P(2 success )
n = 20, p = 0.05, find P(at most 3 successes)

(hint for c.
P (at most 3 successes) = P(x ≤3)= P(x= 0) + P(x = 1)+ P(x = 2)+ P(x = 3)

Answer 1

Using the binomial distribution, it is found that the probabilities are:

• P(X = 4) = 0.1268.

• P(X = 2) = 0.2835.

• $P(X \leq 3) = 0.9841$

What is the binomial distribution formula?

The formula is:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The parameters are:

• x is the number of successes.

• n is the number of trials.

• p is the probability of a success on a single trial.

Exercise 1:

The parameters are:

n = 15, p = 0.4, x = 4.

Then:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 4) = C_{15,4}.(0.4)^{4}.(0.6)^{11} = 0.1228$

Exercise 2:

The parameters are:

n = 12, p = 0.2, x = 2.

Then:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 2) = C_{12,2}.(0.2)^{2}.(0.8)^{10} = 0.2835$

Exercise 3:

The parameters are:

n = 20, p = 0.05.

The probability is:

$P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

Using the binomial formula for each value and adding them:

$P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.3585 + 0.3774 + 0.1887 + 0.0596 = 0.9841$

More can be learned about the binomial distribution at https://brainly.com/question/24863377

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