Question 1119912
.
A quick quiz consists of 3 multiple choice problems,  each of which has  5  answers,  only one of which is correct. 
If you make random guesses on all  3  problems,

What is the probability that all  3  of your answers are incorrect ?
What is the probability that all  3  of your answers are correct ?



<pre>
    Imagine that for each of the 3 multiple choice question the answers are labeled by 5 letters A, B, C, D and E  

    (5 possible answers to each question).

     By answering to each question, you mark your answer by one of the 5 letters.

     So, by answering to 3 questions, you write the word of the length 3, using one of 5 letters in each of the three positions.


     It is your model. The entire space of events consists of all {{{5^3}}} = 125 such words of the length 3, written in 5-letter alphabet.


         Now we are ready to answer the problem's questions.
</pre>


(a)  &nbsp;&nbsp;<U>What is the probability that all &nbsp;3&nbsp; of your answers are incorrect ?</U>


<pre>
     If all 3 of your answers are incorrect, it means that in each of the 3 positions you put one of 4 letters, distinct of correct.

     You can do it in  {{{4^3}}} ways, therefore, the probability under the question is  {{{4^3/5^3}}} = {{{(4/5)^3}}} = {{{0.8^3}}} = 0.512.
</pre>


(b)  &nbsp;&nbsp;<U>What is the probability that all &nbsp;3&nbsp; of your answers are correct ?</U>


<pre>
     If all 3 of your answers are correct, it means that in each of the 3 positions you guessed the correct letter.

     There is <U>ONLY ONE way</U> to do it; therefore, the probability under the question is  {{{1/5^3}}} = {{{1/125}}} = 0.008.
</pre>


All questions are answered.


The problem is solved.