A bag contains 4 red balls, 6 blue balls and 5 green balls. Two balls are drawn randomly without replacement. What is the probability that one ball is red and the other is blue? Are these two events independent?
Answer
Step 5: To check if the events are independent, we need to check if the Probability (Red and Blue) = Probability (Red) x Probability (Blue). If they are equal, the events are independent. If not, they are dependent. In this case, Probability (Blue) = \(\frac{6}{15}\). So, Probability (Red) x Probability (Blue) = \(\frac{4}{15}\) x \(\frac{6}{15}\) = \(\frac{24}{225}\) = \(\frac{8}{75}\). Since \(\frac{4}{35}\) ≠ \(\frac{8}{75}\), the events are dependent.
Step 1 ：Step 1: Calculate the total number of balls. Total number of balls = 4 (red) + 6 (blue) + 5 (green) = 15.
Step 2 ：Step 2: Calculate the probability of drawing a red ball in the first draw. Probability (Red) = \(\frac{4}{15}\).
Step 3 ：Step 3: After drawing a red ball, the total number of balls is now 14. Then, calculate the probability of drawing a blue ball in the second draw. Probability (Blue after Red) = \(\frac{6}{14}\).
Step 4 ：Step 4: Calculate the probability of both events happening. Probability (Red and Blue) = Probability (Red) x Probability (Blue after Red) = \(\frac{4}{15}\) x \(\frac{6}{14}\) = \(\frac{24}{210}\) = \(\frac{8}{70}\) = \(\frac{4}{35}\).
Step 5 ：Step 5: To check if the events are independent, we need to check if the Probability (Red and Blue) = Probability (Red) x Probability (Blue). If they are equal, the events are independent. If not, they are dependent. In this case, Probability (Blue) = \(\frac{6}{15}\). So, Probability (Red) x Probability (Blue) = \(\frac{4}{15}\) x \(\frac{6}{15}\) = \(\frac{24}{225}\) = \(\frac{8}{75}\). Since \(\frac{4}{35}\) ≠ \(\frac{8}{75}\), the events are dependent.
