# Solved (Free): The school finds that 30% of the students are low achievers. The Headmaster is instructed to devise a revision module that will improve the situation

#### ByDr. Raju Chaudhari

Mar 14, 2021

The school finds that 30% of the students are low achievers. The Headmaster is instructed to devise a revision module that will improve the situation. One hundred of the students are chosen at random and are given the revision module. Out of these, 90% of the high achievers pass the test and 20% of the low achievers also pass. Based on these results, if a student chosen at random, takes the module, and passes it, what is the probability that he or she is a high achiever?
If the student fails the module, what is the probability that he or she is a high achiever?

#### Solution

Given that 30% of the students are low achievers, so 70% of the students are high achievers.

Also 90% of the high achievers pass the test, so the number of high achiever students who pass the test is 90% of 70. That is 63 high achiever students pass the test.

And 20% of the low achievers also pass the test, so the number of low achoever students who pass the test equals 20% of 30. That is 6 low achiever students pass the test.

. Pass Fail Total
High Achievers 63 7 70
Low Achievers 6 24 30
Total 69 31 100

If a student chosen at random, takes the module, and passes it, then the probability that he or she is a high achiever is

 \begin{aligned} P(\text{High}|\text{Pass}) &= \frac{P(\text{High}\cap \text{Pass})}{P(\text{Pass})}\\ &=\frac{63/100}{69/100}\\ &=\frac{63}{69}\\ &=0.9130435 \end{aligned}

If the student fails the module, then the probability that he or she is a high achiever is

 \begin{aligned} P(\text{High}|\text{Fail}) &= \frac{P(\text{High}\cap \text{Fail})}{P(\text{Fail})}\\ &=\frac{7/100}{31/100}\\ &=\frac{7}{31}\\ &=0.2258065 \end{aligned}