Question

Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an A grade. What is the probability that the student has 100% attendance? Is regularity required only in school? Justify your answer.

Answer 1

Hint: In simple words, probability is how likely something to happen is. The probability of happening of an event B when an event A has already happened is called conditional probability of event A over B. Use this concept of conditional probability and Bayes theorem to find out the correct answer.

Complete step-by-step answer:
Let E be the event of students that have 100% attendance, $ P(E) = \dfrac{{30}}{{100}} = 0.3 $
Let F be the event of irregular students, $ P(F) = \dfrac{{70}}{{100}} = 0.7 $
Now let A be the event that a student scores an A grade.
We know that out of all the students who have 100% attendance, 70% score A grade, so
The probability of a student getting an A grade given that he has 100% attendance is $ P(A/E) = \dfrac{{70}}{{100}} = 0.7 $
Also, we know that out of all the irregular students, 10% attain A grade, so
The probability of a student getting an A grade given that he doesn’t have 100% attendance is
 $ P(A/F) = \dfrac{{10}}{{100}} = 0.1 $
Now, by Bayes theorem, we know that
$ P({E_i}/A) = \dfrac{{P({E_i}) \cdot P(A/{E_i})}}{{\sum\limits_{i = 1}^n {P({E_i}) \cdot P(A/{E_i})} }}\forall i = 1,2,3,....,n $
So the probability that a student has 100% attendance given that he attained an A grade is –
 $
  P(E/A) = \dfrac{{P(E) \cdot P(A/E)}}{{P(E) \cdot P(A/E) + P(F) \cdot P(A/F)}} \\
  P(E/A) = \dfrac{{0.3 \times 0.7}}{{0.3 \times 0.7 + 0.7 \times 0.1}} = \dfrac{3}{4} = 0.75 \\
  $
No, regularity is required in every field of work because consistency is the key to success. Only those who are dedicated to doing hard work regularly achieve their goals.
So, the correct answer is “0.75”.

Note: Mathematically, probability can be defined as the number of favorable outcomes divided by the total number of possible outcomes. Let n be the total possible outcomes of an event and m of them be the favourable outcomes, then
 $ probability = \dfrac{m}{n} $
Conditional probability is denoted by $ P(A/B) = \dfrac{{P(A \cap B)}}{{P(B)}} $ , where $ P(A \cap B) $ represents the probability of the union of the events A and B and $ P(B) \ne 0 $ .
Similarly, $ P(B/A) = \dfrac{{P(B \cap A)}}{{P(A)}} $ , where $ P(A) \ne 0 $ .