Determining the Probability of Overlapping Preferences in a University Class

In a randomized university class, 60 students expressed a liking for English, and 40 students liked Maths. We are given specific probabilities that a student likes one subject given the other. Using this information, we can determine the probability of a student liking both English and Maths. This example demonstrates the application of conditional probability and set theory in solving practical problems.

Concept of Conditional Probability

The problem at hand can be solved by applying the concept of conditional probability. The steps involved are as follows:

Step 1: Define Probabilities

tPE Probability that a student likes English 0.60 tPM Probability that a student likes Maths 0.40 tPE|M Probability that a student likes English given that they like Maths 0.20

Step 2: Apply the Conditional Probability Formula

The conditional probability formula is:

PE|M PE cap M / PM

Rearranging this formula, we can find the probability that a student likes both English and Maths:

PE cap M PE|M * PM

Step 3: Calculate the Probability

Substituting the known values into the formula:

PE cap M 0.20 * 0.40 0.08

Multiple Verification Methods

The solution can also be verified using multiple methods, such as set theory and Venn diagrams:

Venn Diagram Method

From the Venn diagram:

t9 students like only Maths. t2 students like only English. t10 students like both English and Maths. t4 students do not like either subject.

Therefore, the probability of randomly choosing a student that likes both subjects is:

PE cap M 10 / 25 0.40

Set Theory Verification

Using the inclusion/exclusion formula:

PE cap M ME - M union E 19 / 25 - 25 - 4 10

Thus, the probability that a student likes both subjects is:

PM cap E 10 / 25 0.40

Alternative Method Verification

Another method involves recognizing that 20 out of 40 students who like Maths also like English. This translates to:

20 / 40 0.5 * 40 8

Hence, the probability is 8%, or 0.08, indicating an 8% chance that a student likes both Maths and English.

Conclusion

In summary, using conditional probability and set theory, we determined that the probability of a student liking both English and Maths is 8%. This example illustrates the practical application of mathematical concepts in real-world scenarios, such as analyzing student preferences in a university setting.

By understanding and applying these principles, we can gain valuable insights into overlapping preferences and make informed decisions based on statistical analysis.