Probability of Selecting a Student Who Passed the Exam or Is Male

Introduction

Probability theory is a fundamental concept in mathematics with numerous applications, particularly in fields such as statistics, data science, and artificial intelligence. In this article, we'll explore the scenario of a class of 40 students, where a certain number of males and females have passed their final exam. We'll calculate the probability of selecting a student at random who either passed the exam or is male. This problem involves the principles of probability and the application of the principle of inclusion-exclusion.

Problem Description

In a class of 40 students, there are 18 males and 22 females. Two-thirds of the male students and one-half of the female students passed the final exam. We need to find the probability that a randomly chosen student either passed the exam or is male.

Step-by-Step Solution

Step 1: Determine the number of students who passed the exam.

Male Students: Total male students 18 Passed (frac{2}{3} times 18 12) Female Students: Total female students 22 Passed (frac{1}{2} times 22 11)

Step 2: Calculate the total number of students who passed the exam.

Total students who passed Male passed Female passed 12 11 23

Step 3: Calculate the total number of male students.

Total male students 18

Step 4: Calculate the total number of students in the class.

Total students 40

Step 5: Use the principle of inclusion-exclusion to find the number of students who either passed the exam or are male.

Let:

(A) the event that a student passed the exam (B) the event that a student is male

We need to find (P(A cup B)):

[ P(A cup B) P(A) P(B) - P(A cap B) ]

Step 6: Calculate (P(A)), (P(B)), and (P(A cap B)).

(P(A)) Probability that a student passed the exam (frac{23}{40}) (P(B)) Probability that a student is male (frac{18}{40}) (P(A cap B)) Probability that a student is male and passed the exam (frac{12}{40}) (since 12 males passed)

Now substituting the values:

[ P(A cup B) frac{23}{40} frac{18}{40} - frac{12}{40} ]

Step 7: Calculate the final probability.

[ P(A cup B) frac{23 18 - 12}{40} frac{29}{40} ]

Conclusion

The probability that a randomly chosen student either passed the exam or is male is (frac{29}{40}).

Reflection on the Question

It is indeed interesting to note that the problem centers on evaluating probabilities based on gender. Although the question is straightforward and aligned with the principles of probability, it prompts discussions around inclusivity and gender balance in educational settings. The probability calculated here, (frac{29}{40}), indicates that a significant portion of the class either passed the exam or identifies as male.

Additional Notes

The problem can further be analyzed by breaking down the intersection of passed students and gender. For instance, the probability of selecting a student who is male and passed the exam is (frac{12}{40}), and the probability of selecting a female who passed the exam is (frac{11}{40}). This layered analysis can provide a deeper understanding of the demographics within the class.

Understanding these concepts is crucial for anyone looking to advance in fields that heavily rely on statistical data and probability theory. Whether you're a student, a professional, or a researcher, the ability to interpret and solve probability problems is a valuable skill.