Question

a student ran out of time on a multiple choice exam and randomly guess the answers for two problems each problem have four answer choices ABCD and only one correct answer what is the probability that he answered neither of the problems correctly ​

Answer 1

Answer:

The probability that he answered neither of the problems correctly ​is 0.0625.

Step-by-step explanation:

We are given that a student ran out of time on a multiple-choice exam and randomly guess the answers for two problems each problem have four answer choices ABCD and only one correct answer.

Let X = Number of problems correctly ​answered by a student.

The above situation can be represented through binomial distribution;

$P(X=r)=\binom{n}{r}\times p^{r}\times (1-p)^{n-r};x=0,1,2,3,....$    

where, n = number of trials (samples) taken = 2 problems

           r = number of success = neither of the problems are correct

           p = probability of success which in our question is probability that

                 a student answer correctly, i.e; p = $\frac{1}{4}$ = 0.75.

So, X ~ Binom(n = 2, p = 0.75)

Now, the probability that he answered neither of the problems correctly ​is given by = P(X = 0)

             P(X = 0) = $\binom{2}{0}\times 0.75^{0}\times (1-0.75)^{2-0}$

                            = $1 \times 1\times 0.25^{2}$

                            = 0.0625