Respuesta :
Answer:
[tex] P(X=0)[/tex]
And using the probability mass function we got:
[tex] P(X=0) =(30C0) (0.06)^0 (1-0.06)^{30-0}= 0.156[/tex]
Step-by-step explanation:
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 now that:
[tex]X \sim Binom(n=30, p=0.06)[/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 we want this probability:
[tex] P(X=0)[/tex]
And using the probability mass function we got:
[tex] P(X=0) =(30C0) (0.06)^0 (1-0.06)^{30-0}= 0.156[/tex]
Answer:
The probability that none of the houses will develop a leak is 0.156 (To the nearest thousandths)
Step-by-step explanation:
Given that the probability that a house in an urban area will develop a leak is 6%, this implies that the probability is 6/100 = 0.06
Suppose 30 houses are randomly selected, to find the probability that none of the houses will develop a leak, we make use of the Binomial Distribution,
If x is approximated Binomial(n, p)
Then
P(x) = nCxP^xq^(n - x)
Where q = 1 - p
And nCx, read as "n combination x"
is given as n!/(n - x)! x!
Here, x = 0, n = 30, p = 0.06
P(0) = (30C0)(0.06)^0(0.94)^(30 - 0)
= (30!/(30-0)!0! (1)(0.156)
= 1 × 0.156
= 0.156.
