Question 4 An opaque bag contains 21 yellow marbles and 12 teal marbles. If two marbles are randomly chosen from the bag at the same time, find the probability that both marbles are teal. Round your answer to four decimal places. Submit Question

To find the probability that both marbles drawn are teal, you first need to calculate the total number of marbles in the bag, which is 21 yellow + 12 teal = 33 marbles. When you draw two marbles, the total ways to choose 2 marbles out of 33 is given by the combination formula C(n, k) = n! / (k!(n-k)!), which results in C(33, 2) = 528. Next, to find the number of ways to choose 2 teal marbles from 12, use C(12, 2) = 66. Thus, the probability is 66/528, which simplifies to 0.125 or 0.1250 when rounded to four decimal places. The interesting journey of probability traces back to the 17th century, intertwined with the world of gambling! Mathematicians like Blaise Pascal and Pierre de Fermat laid the groundwork for probability theory while trying to devise fair games. They developed concepts that turned randomness into a calculable science, making it possible to understand and predict outcomes in uncertain scenarios—like randomly drawing marbles from a bag! In real life, understanding probability can help us in decision-making! For instance, if you're a game designer or a statistician, knowing how to calculate odds can help create balanced games or interpret data trends. This knowledge extends beyond games, too—financial analysts use probability to assess risks, and healthcare professionals use it for evaluating treatment outcomes. It’s all around us, helping us make informed choices based on calculated risks.