Respuesta :
Answer:
18.75% probability that more than 3 of them have a high school diploma
Step-by-step explanation:
For each adult worker, there are only two possible outcomes. Either they have a high school diploma, or they do not. The adults are chosen at random, which means that the probability of an adult having a high school diploma is independent from other adults. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
It is known that 50% of adult workers have a high school diploma.
This means that [tex]p = 0.5[/tex]
If a random sample of 5 adult workers is selected, what is the probability that more than 3 of them have a high school diploma?
This is P(X > 3) when [tex]n = 5[/tex]
So
[tex]P(X > 3) = P(X = 4) + P(X = 5)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 4) = C_{5.4}.(0.5)^{4}.(0.5)^{1} = 0.15625[/tex]
[tex]P(X = 5) = C_{5.5}.(0.5)^{5}.(0.5)^{0} = 0.03125[/tex]
[tex]P(X > 3) = P(X = 4) + P(X = 5) = 0.15625 + 0.03125 = 0.1875[/tex]
18.75% probability that more than 3 of them have a high school diploma
Answer:
The probability that high school diploma is possessed by more than 3 is 0.1875.
Step-by-step explanation:
It is given that 50% of adult workers have a high school diploma.
Therefore we can write p = 0.5.
We are given to find that if a random sample of 5 adult workers is selected what is the probability that more than 3 of them have a high school diploma.
Therefore we can say that the sample size, n = 8.
This is a problem that will be solved using Binomial Theorem.
We have to find p(X > 3) which can be written as
p(X > 3) = p(X = 4) + P(X = 5)
= [tex]\binom{8}{4}(0.5)^4(0.5)^1 + \binom{8}{5}(0.5)^5(0.5)^0[/tex]
= (6)([tex]0.5^5[/tex])
= 0.1875
The probability that high school diploma is possessed by more than 3 is 0.1875.
