Respuesta :
Using the binomial distribution, it is found that the probability that at most 4 students have taken chemistry is 0.438.
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 values of the parameters are given by:
p = 0.4, n = 12.
The probability that at most 4 students have taken chemistry is given by:
[tex]P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)[/tex]
In which:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{12,0}.(0.4)^{0}.(0.6)^{12} = 0.002[/tex]
[tex]P(X = 1) = C_{12,1}.(0.4)^{1}.(0.6)^{11} = 0.017[/tex]
[tex]P(X = 2) = C_{12,2}.(0.4)^{2}.(0.6)^{10} = 0.063[/tex]
[tex]P(X = 3) = C_{12,3}.(0.4)^{3}.(0.6)^{9} = 0.142[/tex]
[tex]P(X = 4) = C_{12,4}.(0.4)^{4}.(0.6)^{8} = 0.213[/tex]
Then:
[tex]P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 0.002 + 0.017 + 0.063 + 0.142 + 0.213 = 0.438[/tex]
More can be learned about the binomial distribution at https://brainly.com/question/24863377
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