Using the binomial distribution, it is found that the probability that he will hit fewer than 8 of the bottles is of 0.012, given by option A.
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:
In this problem, we have that:
The probability that he hits fewer than 8 of the bottles is given by:
[tex]P(X < 8) = 1 - P(X \geq 8)[/tex]
In which:
[tex]P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10)[/tex]
Then:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 8) = C_{10,8}.(0.95)^{8}.(0.05)^{2} = 0.0746[/tex]
[tex]P(X = 9) = C_{10,9}.(0.95)^{9}.(0.05)^{1} = 0.3151[/tex]
[tex]P(X = 10) = C_{10,10}.(0.95)^{10}.(0.05)^{0} = 0.5987[/tex]
Hence:
[tex]P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) = 0.0746 + 0.3151 + 0.5987 = 0.988[/tex]
[tex]P(X < 8) = 1 - P(X \geq 8) = 0.012[/tex]
The probability that he will hit fewer than 8 of the bottles is of 0.012, given by option A.
More can be learned about binomial distribution at https://brainly.com/question/24863377
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