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Van guessed on all 8 questions of a multiple-choice quiz. Each question has 4 answer choices. What is the probability that he got exactly 1 question correct? Round the answer to the nearest thousandth. P (k successes) = Subscript n Baseline C Subscript k Baseline p Superscript k Baseline (1 minus p) Superscript n minus k. Subscript n Baseline C Subscript k Baseline = StartFraction n factorial Over (n minus k) factorial times k factorial EndFraction 0.033 0.267 0.461 0.733

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musiclover10045

4 answer choices with one correct answer.

The probability of getting the correct answer on a question would be 1/4

The probability of getting a wrong answer would be 3/4

The overall probability of getting just 1 right is, The probability the first is right and the rest are wrong + the probability the second answer is correct and the rest are wrong, etc.

8 x 1/4 x 3/4 x 3/4 x 3/4 x 3/4 x 3/4 x 3/4 x 3/4 = 0.267

Using the binomial distribution, it is found that there is a 0.267 probability that he got exactly 1 question correct.

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:

  • Each question has one correct option out of four, and he guesses, hence p = 1/4 = 0.25.
  • There is a total of 8 questions, hence n = 8.

The probability that he got exactly 1 question correct is P(X = 1), hence:

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 1) = C_{8,1}.(0.25)^{1}.(0.75)^{7} = 0.267[/tex]

More can be learned about the binomial distribution at https://brainly.com/question/24863377