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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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