Respuesta :
Answer:
0.649 = 64.9% probability that he will answer no more than 3 questions correctly.
Step-by-step explanation:
For each question, there are only two possible outcomes. Either Richard guesses the correct answer, or he does not. The probability of Richard guessing the correct answer in a question is independent of any 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]
Each one with four options, one of which is correct.
This means that [tex]p = \frac{1}[4} = 0.25[/tex]
Assuming that Richard guesses on all 12 questions, find the probability that he will answer no more than 3 questions correctly.
[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{12,0}.(0.25)^{0}.(0.75)^{12} = 0.032[/tex]
[tex]P(X = 1) = C_{12,1}.(0.25)^{1}.(0.75)^{11} = 0.127[/tex]
[tex]P(X = 2) = C_{12,2}.(0.25)^{2}.(0.75)^{10} = 0.232[/tex]
[tex]P(X = 3) = C_{12,3}.(0.25)^{3}.(0.75)^{9} = 0.258[/tex]
[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.032 + 0.127 + 0.232 + 0.258 = 0.649[/tex]
0.649 = 64.9% probability that he will answer no more than 3 questions correctly.
