Answer :
Answer:
a) p=0.2
b) probability of passing is 0.01696
.
c) The expected value of correct questions is 1.2
Step-by-step explanation:
a) Since each question has 5 options, all of them equally likely, and only one correct answer, then the probability of having a correct answer is 1/5 = 0.2.
b) Let X be the number of correct answers. We will model this situation by considering X as a binomial random variable with a success probability of p=0.2 and having n=6 samples. We have the following for k=0,1,2,3,4,5,6
[tex] P(X=k) = \binom{n}{k}p^{k}(1-p)^{n-k} = \binom{6}{k}0.2^{k}(0.8)^{6-k}[/tex].
Recall that [tex]\binom{n}{k}= \frac{n!}{k!(n-k)!}[/tex] In this case, the student passes if X is at least four correct questions, then
[tex]P(X\geq 4) = P(X=4)+P(X=5)+P(X=6)=\binom{6}{4}0.2^{4}(0.8)^{6-4}+\binom{6}{5}0.2^{5}(0.8)^{6-5}+\binom{6}{6}0.2^{6}(0.8)^{6-6}= 0.01696 [/tex]
c)The expected value of a binomial random variable with parameters n and p is [tex]E[X] = np[/tex]. IN our case, n=6 and p =0.2. Then the expected value of correct answers is [tex]6\cdot 0.2 = 1.2[/tex]