Respuesta :
Using the binomial distribution, it is found that the probabilities are given as follows:
a) 0.3185 = 31.85%.
b) 0.7998 = 79.98%.
c) 0.5187 = 51.87%.
What is the binomial distribution formula?
The formula is:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
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.
In this problem, the parameters are given as follows:
n = 5, p = 0.51.
Item a:
The probability is P(X = 3), hence:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 3) = C_{5,3}.(0.51)^{3}.(0.49)^{2} = 0.3185[/tex]
Item b:
The probability is:
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
Then:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{5,0}.(0.51)^{0}.(0.49)^{5} = 0.0282[/tex]
[tex]P(X = 1) = C_{5,1}.(0.51)^{1}.(0.49)^{4} = 0.1470[/tex]
[tex]P(X = 2) = C_{5,2}.(0.51)^{2}.(0.49)^{3} = 0.3061[/tex]
[tex]P(X = 3) = C_{5,3}.(0.51)^{3}.(0.49)^{2} = 0.3185[/tex]
So:
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.0282 + 0.1470 + 0.3061 + 0.3185 = 0.7998.
Item c:
The probability is:
[tex]P(X \geq 3) = 1 - P(X < 3)[/tex]
In which:
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.0282 + 0.1470 + 0.3061 = 0.4813.
Then:
[tex]P(X \geq 3) = 1 - P(X < 3) = 1 - 0.4813 = 0.5187[/tex]
More can be learned about the binomial distribution at https://brainly.com/question/24863377
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