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A test has 20 true/false questions. What is the probability that a student passes the test if they guess the answers? Passing means that the student needs to get 15 questions correct. (Hint: you must determine the probability of success; look at the number of choices of answers the student will select from) The probability that the student will get 15 correct questions in this test by guessing is

Sagot :

Using the binomial distribution, it is found that:

The probability that the student will get 15 correct questions in this test by guessing is 0.0207 = 2.07%.

For each question, there are only two possible outcomes, either the guess is correct, or it is not. The guess on a question is independent of any other question, hence, the binomial distribution is used to solve this question.

Binomial probability distribution

[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:

  • There are 20 questions, hence [tex]n = 20[/tex].
  • Each question has 2 options, one of which is correct, hence [tex]p = \frac{1}{2} = 0.5[/tex]

The probability is:

[tex]P(X \geq 15) = P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20)[/tex]

In which:

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

[tex]P(X = 15) = C_{20,15}.(0.5)^{15}.(0.5)^{5} = 0.0148[/tex]

[tex]P(X = 16) = C_{20,16}.(0.5)^{16}.(0.5)^{4} = 0.0046[/tex]

[tex]P(X = 17) = C_{20,17}.(0.5)^{17}.(0.5)^{3} = 0.0011[/tex]

[tex]P(X = 18) = C_{20,18}.(0.5)^{18}.(0.5)^{2} = 0.0002[/tex]

[tex]P(X = 16) = C_{20,19}.(0.5)^{19}.(0.5)^{1} = 0[/tex]

[tex]P(X = 17) = C_{20,20}.(0.5)^{20}.(0.5)^{0} = 0[/tex]

Then:

[tex]P(X \geq 15) = P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20) = 0.0148 + 0.0046 + 0.0011 + 0.0002 + 0 + 0 = 0.0207[/tex]

The probability that the student will get 15 correct questions in this test by guessing is 0.0207 = 2.07%.

You can learn more about the binomial distribution at https://brainly.com/question/24863377

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