Answer:
The probability that the student answers all questions incorrectly is 0.1074
The probability that the student will achieve at least 50% correct is 0.0328
Step-by-step explanation:
This exercise adjust to a normal distribution, where:
p: probability that the student answers the question correctly ([tex]\frac{1}{5}[/tex])
n: number of questions (10 questions)
The binomial distribution is given by:
[tex]P(X=x)=\frac{n!}{x!(n-x)!}\times p^x \times (1-p)^{n-x}[/tex]
The probability that the student answers all questions incorrectly is P(X=0)
[tex]P(X=0)=\frac{10!}{0!(10-0)!}\times (0.2)^0 \times (1-0.2)^{10-0}=0.1074[/tex]
The probability that the student will achieve at least 50% correct is P(X≥5)
P(X≥5)= 1 - P(X=0) - P(X=1) - P(X=2) - P(X=3) - P(X=4)
P(X=0)=0.1074
[tex]P(X=1)=\frac{10!}{1!(10-1)!}\times (0.2)^1 \times (1-0.2)^{10-1}=0.2684[/tex]
[tex]P(X=2)=\frac{10!}{2!(10-2)!}\times (0.2)^2 \times (1-0.2)^{10-2}=0.3020[/tex]
[tex]P(X=3)=\frac{10!}{3!(10-3)!}\times (0.2)^3 \times (1-0.2)^{10-3}=0.2013[/tex]
[tex]P(X=4)=\frac{10!}{4!(10-4)!}\times (0.2)^4 \times (1-0.2)^{10-4}=0.0881[/tex]
P(X≥5)= 1 - 0.1074 - 0.2684 - 0.3020 - 0.2013 - 0.0881=0.0328
