Respuesta :
Answer:
The probability that the student answers at least seventeen questions correctly is [tex]8.03\times 10^{-10}[/tex].
Step-by-step explanation:
Let the random variable X represent the number of correctly answered questions.
It is provided all the questions have five options with only one correct option.
Then the probability of selecting the correct option is,
[tex]P(X)=p=\frac{1}{5}=0.20[/tex]
There are n = 20 question in the exam.
It is also provided that a student taking the examination answers each of the questions with an independent random guess.
Then the random variable can be modeled by the Binomial distribution with parameters n = 20 and p = 0.20.
The probability mass function of X is:
[tex]P(X=x)={20\choose x}\ 0.20^{x}\ (1-0.20)^{20-x};\ x =0,1,2,3...[/tex]
Compute the probability that the student answers at least seventeen questions correctly as follows:
[tex]P(X\geq 17)=P (X=17)+P (X=18)+P (X=19)+P (X=20)[/tex]
[tex]=\sum\limits^{20}_{x=17}{{20\choose x}\ 0.20^{x}\ (1-0.20)^{20-x}}\\\\=0.00000000077+0.000000000032+0.00000000000084+0.000000000000042\\\\=0.000000000802882\\\\=8.03\times10^{-10}[/tex]
Thus, the probability that the student answers at least seventeen questions correctly is [tex]8.03\times 10^{-10}[/tex].
