Question

Five multiple choice questions containing 5 options each are to be answered by a student.If the pass mark of the test is 60%, what is the probability that the student fails the test?

Answer 1

In this problem, a student is doing a test:

• with five multiple-choice questions → n = 5,

,

• containing five options each → probability of guess p = 1/5 = 0.2,

,

• a pass mark of the test of 60%.

We want to know the probability that the student fails the test, i.e. we must compute the probability:

`P_(n=5,p=0.2)(X<3).`

Where X is the number of correct answers and X = 3 = 0.6 * 5 represents a 60% of correct answers in the test.

Now, this probability consists of a binomial probability, which refers to the probability of exactly X successes on n repeated trials in an experiment that has two possible outcomes.

The formula for the binomial probability is:

`P_(n,p)(X)=(n!)/(X!\cdot(n-X)!)\cdot p^X\cdot(1-p)^(n-X).`

Now, the probability above is given by:

`P_(5,0.2)(X<3)=P_(5,0.2)(X=0)+P_(5,0.2)(X=1)+P_(5,0.2)(X=2)\text{.}`

Using the formula above, we compute each term, we get:

`\begin{gathered} P_(5,0.2)(X=0)=(5!)/(0!\cdot(5-0)!)\cdot0.2^0\cdot(1-0.2)^(5-0)=0.32768, \\ P_(5,0.2)(X=1)=(5!)/(1!\cdot(5-1)!)\cdot0.2^1\cdot(1-0.2)^(5-1)=0.4096, \\ P_(5,0.2)(X=2)=(5!)/(2!\cdot(5-2)!)\cdot0.2^2\cdot(1-0.2)^(5-2)=0.2048. \end{gathered}`

Replacing these values in the formula above, we get:

`P_(5,0.2)(X<3)=0.94208.`

Answer

The probability that the student fails the test is 0.94208.