Question

There are 5 brilliant students in class XI and 8 brilliant student is class XII each class has 50 students, the odds in favor of choosing the class XI are 2:3, If the class XI is not chosen then the class XII is chosen, A student is a chosen and is found to be brilliant, find the probability that the chosen student is from class XI.

Answer 1

Hint: This is a question of conditional probability. Try recalling the concepts and formula of conditional probability. The basic formula is ${\text{P(A/B)}}\,{\text{ = }}\,\dfrac{{{\text{P(A}} \cap {\text{B)}}}}{{{\text{P(B)}}}}$ .

Complete step-by-step answer:
Let’s consider ‘E’ and ‘F’ events of ‘class XI is chosen’ and class XII is chosen respectively.
Then, according to the given ration in the question.
The probability of choosing a brilliant student from class XI us, ${\text{P(E)}}\,{\text{ = }}\,\dfrac{{\text{2}}}{{\text{5}}}$ .
The probability of choosing a brilliant student from class XII is, ${\text{P(F)}}\,{\text{ = }}\,\dfrac{3}{{\text{5}}}$
Let us consider ‘A’ to be the event for the student chosen is brilliant.
The formula for P(A) is,
$\therefore $ P(A) = P(E)$ \times $ P(A/E) + P(F) $ \times $ P(A/F)
Substituting the volume,
\[{\text{P(A)}}\,{\text{ = }}\,\dfrac{{\text{2}}}{{\text{5}}}{{ \times }}\,\dfrac{{\text{5}}}{{{\text{50}}}}\,{\text{ + }}\,\dfrac{{\text{3}}}{{\text{5}}}\,{{ \times }}\,\dfrac{{\text{8}}}{{{\text{50}}}}\,{\text{ = }}\,\dfrac{{{\text{34}}}}{{{\text{250}}}}\]
Now, the probability that the chosen student is from class XI is,
\[
  {\text{P(E/A)}}\,{\text{ = }}\,\dfrac{{{\text{P(E)}}{\text{.P(E/A)}}}}{{{\text{P(E)}}{\text{.P(A/E) + P(F)}}{\text{.P(A/F)}}}} \\
  {\text{ = }}\,\dfrac{{{\text{2/}}{{\text{5}}}\,{{ \times }}\,{\text{5/}}{{{\text{50}}}}}}{{{\text{34/}}{{{\text{250}}}}}}\,{\text{ = }}\,\dfrac{{{{\text{2}}}{{ \times }}\,{\text{5}}}}{{{{{\text{}}}}{\text{34}}}}\,{\text{ = }}\,\dfrac{{\text{5}}}{{{\text{17}}}} \\
\]
The probability of the student chosen is from class XI is 5/17.

Note: Remember the basic concepts of probability and always assign alphabets to the events so that it’ll become easy to use them into formula.