Understanding conditional probability in a problem

Question

The problem is as follows:

Senior students tend to stay up all night and therefore are not able to wake up on time in morning. Not only this but their dependence on tuitions further leads to absenteeism in school. Of the students in class XII, it is known that 30% of the students have 100% attendance. Previous year results report that 80% 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 class XII.

Find the conditional probability that a student attains A grade given that he is not 100 % regular student.

My working:

I made a map of how the events occur below, and with that, it can be inferred that the probability 'Grade A' obtained by an irregular student is:

$$ = P(being \ irregular)P(Grade A)$$ $$ = \frac{7}{10} \frac{1}{10} $$

However, the answer given in my book is $\frac{1}{10} $.

I don't really get any idea of how it is possible. From a group of 100 students, it is probable that 70 of them are irregular and among them, $70 \frac{1}{10}$ - which is 7 students out of the hundred.

But on deeper thought, since the problems ask the probability of getting grade A among the irregulars, the probability becomes $\frac{7}{70}$.

Now my question is, what should the question be, such that i get the answer - $\frac{7}{10} \frac{1}{10}$ ?

Answer 1

To get the answer $\frac{7}{10}\cdot\frac{1}{10}$, the new question can be: "What is the probability that the randomly chosen student is an irregular student, who scores an A?".

In this case you would have these calculations $$P(\text{Grade A and No 100%})=P(\text{No 100%})\cdot P(\text{Grade A}|\text{No 100%})=\frac{7}{10}\cdot\frac{1}{10}$$ ("$\text{Grade A}|\text{No 100%}$" reads as "Grade A, given No $100\%$")

The original question was asking for the conditional probability. It was basically asking "What fraction of irregular students scores an A?".

This is why you would have these calculations and the textbook's answer $$P(\text{Grade A}|\text{No 100%})=\frac{P(\text{Grade A and No 100%})}{P(\text{No 100%})}=\frac{\frac{7}{10}\cdot\frac{1}{10}}{\frac{7}{10}}=\frac{1}{10}$$

Hope this helps and feel free to ask further questions, if this is a bit unclear! :)

Answer 2

The problem is trivial - it gives you that if a student does not have perfect attendance, the probability that they make an A is $10\% = \frac{1}{10}$.

If you want to use the definition of conditional probability here, you can first calculate the probability that a student does not have perfect attendance, which is $1 - 30\% = 70\% = \frac{7}{10}$. Then, the probability that a random student does not have perfect attendance and still makes an A is $10\% (1 - 30\%) = 7\% = \frac{7}{100}$. By Bayes, the probability that a student makes A given that he or she does not have perfect attendance is $\frac{\frac{7}{100}}{\frac{7}{10}} = \frac{1}{10}$.

To answer your question, the answer to

What is the probability that a randomly-selected student (from the general senior population) does not perfect attendance and makes an A?

would be $\frac{7}{100}$.