In a second grade class containing 12 girls and 9 boys, 2 students are selected at random to give out the math papers. What is the probability that both are girls? \( \frac{4}{7} \cdot \frac{4}{7} \) \( \frac{4}{7} \cdot \frac{11}{20} \) \( \frac{4}{7} \cdot \frac{3}{5} \) \( \frac{3}{7} \cdot \frac{2}{5} \)

To solve for the probability that both selected students are girls, first, we need to determine the total number of students, which is \(12 + 9 = 21\). The probability of selecting the first girl is \( \frac{12}{21} \). After selecting one girl, there are now 11 girls left and a total of 20 students remaining. Hence, the probability of selecting a second girl is \( \frac{11}{20} \). Now, multiplying these probabilities gives us: \[ P(\text{both are girls}) = \frac{12}{21} \cdot \frac{11}{20} = \frac{12 \cdot 11}{21 \cdot 20} = \frac{132}{420} = \frac{11}{35}. \] So, we can confirm that our calculations are accurate. In the classroom setting, you might find it interesting that probability plays a significant role in daily life, such as predicting outcomes in games or giving chances of winning. Understanding the basic principles can help students think critically in situations involving chance, like card games or rolling dice! Also, a fun fact about probabilities is that they can be misinterpreted. Many people believe that if something hasn’t happened in a while (like a coin landing heads multiple times), it’s "due" to happen. This misconception is known as the gambler's fallacy! Recognizing these common mistakes can help improve your understanding of real-world applications of probability!