Question

Bag 1 contains 6 red cubes and 10 blue cubes. Bag 2 contains 7 red cubes and 3 blue cubes. Two cubes are drawn at random, the first from bag 1 and the second from bag 2. (a) Find the probability that the cubes are of the same color. (b) Given that the cubes selected are of different colors, find the probability that the red cube was selected from bag 1. Options: Option 1: For (a) 1/3, For (b) 3/10 Option 2: For (a) 1/2, For (b) 7/16 Option 3: For (a) 3/8, For (b) 5/12 Option 4: For (a) 2/5, For (b) 7/15

Answer 1

The probability of drawing cubes of the same color from two different bags is 9/20, and given that the cubes drawn are of different colors, the probability that the red cube was selected from bag 1 is 9/44. None of the provided options match these calculated probabilities.

Step-by-step explanation:

Probability of Same and Different Colored Cubes

The problem stated involves calculating the probability of drawing cubes of the same color from two different bags and a conditional probability regarding different colored cube selections.

For part (a), we calculate the probability that the cubes drawn are of the same color. This can happen in two ways: both cubes are red or both are blue. The probability of both being red is the product of the probabilities of drawing a red cube from each bag, which is (6/16) × (7/10). The probability of both being blue is (10/16) × (3/10). Adding these probabilities gives us the probability of the cubes being of the same color.

For part (b), given that the cubes are of different colors, we need to find the probability that the red cube was drawn from bag 1. This involves calculating the conditional probability of drawing a red cube from bag 1 and a blue cube from bag 2, and then dividing by the probability of drawing cubes of different colors.

Calculations:

• Both cubes are red: (6/16) × (7/10) = 42/160
• Both cubes are blue: (10/16) × (3/10) = 30/160
• Probability of same color: 42/160 + 30/160 = 72/160 = 9/20
• Probability of different colors: 1 - (Probability of same color) = 1 - 9/20 = 11/20
• Probability that the red cube is from bag 1, given different colors: (Probability of red from bag 1 and blue from bag 2)/ (Probability of different colors) = ((6/16) × (3/10)) / (11/20) = 18/160 / 11/20 = 18/160 × 20/11 = 360/1760 = 45/220 = 9/44

Therefore, the probability of drawing cubes of the same color is 9/20 and, given cubes of different colors, the probability that the red cube was drawn from bag 1 is 9/44.

Hence, the correct answer is:

• For (a): 9/20
• For (b): 9/44

This corresponds to none of the provided options, indicating a possible error in the options or the calculations.