Respuesta :
Answer:
0.042% probability she will guess at least 8 correct
Step-by-step explanation:
For each question, there are only two possible outcomes. Either the student guesses it correctly, or he does not. The probability of a student guessing a question correctly is independent of other questions. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
Each question has four alternatives.
One is correct, so [tex]p = \frac{1}{4} = 0.25[/tex]
If the student guesses on 10 of the questions, what is the probability she will guess at least 8 correct
This is [tex]P(X \geq 8)[/tex] when [tex]n = 10[/tex]. So
[tex]P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10)[/tex]
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 8) = C_{10,8}.(0.25)^{8}.(0.75)^{2} = 0.0004[/tex]
[tex]P(X = 9) = C_{10,9}.(0.25)^{9}.(0.75)^{1} = 0.00002[/tex]
[tex]P(X = 10) = C_{10,10}.(0.25)^{10}.(0.75)^{0} \cong 0 [/tex]
[tex]P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) = 0.0004 + 0.00002 = 0.00042[/tex]
0.042% probability she will guess at least 8 correct
