A bag contains one blue, one yellow, and two red marbles. A second bag contains one red, one blue, and two yellow marbles. A random marble from each bag is removed. What is the probability of removing a red and a yellow? Give your answer as a fraction.
Answer (2)
Calculate the probability of drawing a red marble from Bag 1 and a yellow marble from Bag 2: 2 1 × 2 1 = 4 1 .
Calculate the probability of drawing a yellow marble from Bag 1 and a red marble from Bag 2: 4 1 × 4 1 = 16 1 .
Add the two probabilities: 4 1 + 16 1 = 16 5 .
The probability of removing a red and a yellow marble is 16 5 .
Explanation
Analyze the problem Let's analyze the problem. We have two bags of marbles, and we want to find the probability of drawing a red marble from the first bag and a yellow marble from the second bag OR drawing a yellow marble from the first bag and a red marble from the second bag.
Calculate P(Red from Bag 1 and Yellow from Bag 2) First, let's calculate the probability of drawing a red marble from Bag 1 and a yellow marble from Bag 2. Bag 1 has 1 blue, 1 yellow, and 2 red marbles, so the probability of drawing a red marble from Bag 1 is 4 2 = 2 1 . Bag 2 has 1 red, 1 blue, and 2 yellow marbles, so the probability of drawing a yellow marble from Bag 2 is 4 2 = 2 1 . Therefore, the probability of drawing a red marble from Bag 1 and a yellow marble from Bag 2 is 2 1 × 2 1 = 4 1 .
Calculate P(Yellow from Bag 1 and Red from Bag 2) Next, let's calculate the probability of drawing a yellow marble from Bag 1 and a red marble from Bag 2. The probability of drawing a yellow marble from Bag 1 is 4 1 . The probability of drawing a red marble from Bag 2 is 4 1 . Therefore, the probability of drawing a yellow marble from Bag 1 and a red marble from Bag 2 is 4 1 × 4 1 = 16 1 .
Add the probabilities Finally, we add the two probabilities to find the total probability of drawing a red and a yellow marble. So, the total probability is 4 1 + 16 1 = 16 4 + 16 1 = 16 5 .
Examples
This type of probability calculation is useful in scenarios like quality control in manufacturing, where you want to determine the likelihood of selecting specific items from different batches. For instance, if you're inspecting products from two different suppliers, you might want to know the probability of selecting a defective item from one supplier and a high-quality item from the other. This helps in assessing the overall reliability of your supply chain.
The probability of removing a red marble from one bag and a yellow marble from the other is 16 5 . This is calculated by finding the probabilities of each possible scenario (red from Bag 1 and yellow from Bag 2, and yellow from Bag 1 and red from Bag 2) and then adding them together. Therefore, the answer is 16 5 .
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