In a class 60 percent are female, 50 percent have long hair and 45 percent are both. Are being female and having long hair independent?

1. The product test multiplies 0.60 by 0.50, which gives 0.30. 2. The stated joint probability is 0.45, which does not equal 0.30. 3. So the events of being female and having long hair are not independent. 4. The conditional test agrees: 75 percent of the female students have long hair against 50 percent of all students. 5. Knowing that a student is female therefore changes the chance that the student has long hair. 6. A larger joint probability than the product signals a positive association, not independence. _Source: OpenStax Introductory Business Statistics (CC BY 4.0), Ch 3 "Probability Topics", section 3.2 Independent and Mutually Exclusive Events_