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

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

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators