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What is the probability that he will hit fewer than 8 of the bottles?

What Is The Probability That He Will Hit Fewer Than 8 Of The Bottles class=

Sagot :

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.

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, we have that:

  • The probability of a bottle being hit is of p = 0.95.
  • There are n = 10 bottles.

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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