It was found in a college survey of Janta College that \( 60 \% \) students stay as Paying Guests (PGs) while the remaining are day scholars.

Question

CASE STUDY 

It was found in a college survey of Janta College that \( 60 \% \) students stay as Paying Guests (PGs) while the remaining are day scholars. Over the last few years, it was reported that \( 20 \% \) PGs get a distinction while \( 30 \% \) day scholars are able to achieve distinction marks in the year end exams. For a randomly chosen student, what is the probability of: 

(i) achieving distinction marks? 

(ii) student being a day scholar given that she achieved distinction marks?

Answer 1

Let A be the event that students stay as paying guests. 

And B be the event that students are day scholars.

∴ P (A) = 60% = \(\frac {60}{100} = \frac {3}{5}\)

& P (B) = 40 %   = \(\frac {40}{100} = \frac {2}{5}\) 

Let event E be the event that students got distinction 

∴ P \((\frac EA)\) = 20% \(\frac {20}{100} = \frac {1}{5}\) 

&  ∴ P \((\frac EB)\) = 30% \(\frac {30}{100} = \frac {3}{10}\) 

(i) P (E) = P \((\frac EA)\) P(A) + P \((\frac EB)\) P(B)

= \(\frac 15 \times \frac 35 + \frac 3{10} \times \frac 25 = \frac {3}{25} + \frac {3}{25} = \frac {6}{25}\) 

∴ Probability that students achieving distinction marks is \(\frac 6{25}\)

(ii) P (student being a day scholar given that she achieved distinction marks)

= P\((\frac BE) = \frac {P(\frac EB) P(B)}{P(E)}\) 

= \(\frac {\frac {3}{10}\times \frac 25}{\frac {6}{25}} = \frac {\frac 3{25}}{\frac {6}{25}} = \frac {3}{6} = \frac 12\)