Question

An instructor has a question bank consisting of $300$ easy true/false question, $200$ difficult true/false questions, $500$ easy multiple choice questions and $400$ difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?

Answer 1

Hint: We will calculate the sum of true/false questions. Further we will calculate the sum of multiple choice questions. Thereafter, we will find the probability of the sum of true/false questions and sum of multiple choice questions. Further, we will find required probability by using the formula: $P(E) = \dfrac{{favourable\,\,outcomes}}{{total\,\,number\,\,of\,outcomes}}$


Complete step by step solution:
Consider the following events:
Let $A = $easy true/false questions
$B = $difficult true/false questions
\[C = \]difficult multiple choice questions
Then, $A = 300,\,\,B = 200,\,\,C = 500\,\,and\,\,D = 400$
So, total number of questions $ = 300 + 200 + 500 + 400$
Total number of questions $ = 1400$
Now, let number of true/false question $A + B$
Total number of true/false questions $ = 300 + 200$
Total number of true/false questions $ = 500$
Total number of multiple choice questions $ = C + D$
Total number of multiple choice questions $ = 500 + 400$
Total number of multiple choice questions $ = 900$
P(true/false question) $(E) = \dfrac{{favourable\,\,outcomes}}{{total\,\,number\,\,of\,outcomes}}$
Favourable outcomes $500$
Total number outcomes $ = 1400$
Total number outcomes $ = \dfrac{{500}}{{1400}}$
Total number outcomes $ = \dfrac{{500}}{{1400}}$
Total number outcomes $ = \dfrac{5}{4}$
P(multiple choice questions $(E) = \dfrac{{favourable\,\,outcomes}}{{total\,\,number\,\,of\,outcomes}}$
Favourable outcomes $900$
Total number outcomes $ = 1400$
P(multiple choice outcomes) $ = \dfrac{{900}}{{1400}}$
P(multiple choice outcomes) $ = \dfrac{9}{{14}}$
Then, required probability $ = \dfrac{{P(E)}}{{P(E)}}$
Required probability $ = \dfrac{5}{{\dfrac{{\dfrac{{14}}{9}}}{{14}}}}$
Required probability $ = \dfrac{5}{{14}} \times \dfrac{{14}}{9}$
Required probability $ = \dfrac{5}{9}$


Note: Students keep in mind that we should add the value of true/false questions and multiple choice questions separately for calculating the required probability.