Respuesta :
Answer:
D. Find the cumulative probability for 8 in a binomial distribution with n = 20 and p = 0.5.
Step-by-step explanation:
For each question, there are only two possible outcomes. Either the student guesses the correct answer, or he does not. The probability of the student is guessing the answer of 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.
20 questions
This means that [tex]n = 20.
True/false:
For each of them there are 2 answers, one of which is correct, so [tex]p = \frac{1}{2} = 0.5[/tex]
How would you find the probability that the student will get 8 or fewer answers correct?
[tex]P(X \leq 8) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)[/tex]
That is, it is the same as the cumulative probability for 8.
So the correct answer is:
D. Find the cumulative probability for 8 in a binomial distribution with n = 20 and p = 0.5.
To obtain the probability that a student gets 8 or fewer answers correct, we find the cumulative probability for 8 in a binomial distribution with n = 20 and p = 0.5.
Recall :
- [tex] P(x = x) = nCx \times p^{x} * q^{n-x} [/tex]
- n = number of trials = 20
- p = 0.5
- q = 1 - p = 1 - 0.5 = 0.5
[tex] P(x ≤ 8) = p(x=0) + p(x = 1) + p(x = 2) +... + p(x = 8)[/tex]
Using a binomial probability calculator :
[tex] P(x ≤ 8) = p(x=0) + p(x = 1) + p(x = 2) +... + p(x = 8)[/tex]
Hence, the Cummulative probability is the probability for 8 is the total sum of the probabilities of x = 8 or less in the experiment.
Therefore, the appropriate option is D.
Learn more :https://brainly.com/question/12474772
