Respuesta :
Answer:
0.6127 = 61.27% probability of guessing 6 or more questions correctly.
Step-by-step explanation:
For each question, there are only two possible outcomes. Either you guess the correct answer, or you do not. The probability of guessing the correct answer of a question 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.
12 questions:
This means that [tex]n = 12[/tex]
True-false:
Two options, one of which is correct. So [tex]p = \frac{1}[2} = 0.5[/tex]
Find the probability of guessing 6 or more questions correctly.
[tex]P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 6) = C_{12,6}.(0.5)^{6}.(0.5)^{6} = 0.2256[/tex]
[tex]P(X = 7) = C_{12,7}.(0.5)^{7}.(0.5)^{5} = 0.1934[/tex]
[tex]P(X = 8) = C_{12,8}.(0.5)^{8}.(0.5)^{4} = 0.1208[/tex]
[tex]P(X = 9) = C_{12,9}.(0.5)^{9}.(0.5)^{3} = 0.0537[/tex]
[tex]P(X = 10) = C_{12,10}.(0.5)^{10}.(0.5)^{2} = 0.0161[/tex]
[tex]P(X = 11) = C_{12,11}.(0.5)^{11}.(0.5)^{1} = 0.0029[/tex]
[tex]P(X = 12) = C_{12,12}.(0.5)^{12}.(0.5)^{0} = 0.0002[/tex]
[tex]P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) = 0.2256 + 0.1934 + 0.1208 + 0.0537 + 0.0161 + 0.0029 + 0.0002 = 0.6127[/tex]
0.6127 = 61.27% probability of guessing 6 or more questions correctly.
