To solve the problem of finding the probability of picking one black ball and one red ball from a bag containing 5 black balls and 3 red balls, we will follow a systematic approach. ### Step 1: Understand the Total Number of Balls First, we need to determine the total number of balls in the bag: - Black balls: 5 - Red balls: 3 - Total balls = 5 (black) + 3 (red) = 8 balls ### Step 2: Calculate the Total Ways to Pick 2 Balls Next, we calculate the total number of ways to pick 2 balls from the 8 balls. This can be done using the combination formula: \[ \text{Number of ways to choose } r \text{ items from } n \text{ items} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \] For our case, we want to choose 2 balls from 8: \[ \binom{8}{2} = \frac{8!}{2!(8-2)!} = \frac{8 \times 7}{2 \times 1} = 28 \] So, there are 28 different ways to pick any 2 balls from the bag. ### Step 3: Calculate the Favorable Outcomes Now, we need to find the number of favorable outcomes where one ball is black and the other is red. There are two scenarios for picking one black and one red ball: 1. Pick a black ball first and then a red ball. 2. Pick a red ball first and then a black ball. #### Scenario 1: Black first, then Red - The number of ways to choose 1 black ball from 5: \(\binom{5}{1} = 5\) - The number of ways to choose 1 red ball from 3: \(\binom{3}{1} = 3\) - Total ways for this scenario: \(5 \times 3 = 15\) #### Scenario 2: Red first, then Black - The number of ways to choose 1 red ball from 3: \(\binom{3}{1} = 3\) - The number of ways to choose 1 black ball from 5: \(\binom{5}{1} = 5\) - Total ways for this scenario: \(3 \times 5 = 15\) ### Step 4: Combine the Favorable Outcomes Now, we add the favorable outcomes from both scenarios: \[ \text{Total favorable outcomes} = 15 + 15 = 30 \] ### Step 5: Calculate the Probability The probability of picking one black ball and one red ball is given by the ratio of the number of favorable outcomes to the total outcomes: \[ P(\text{1 black and 1 red}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{30}{28} = \frac{15}{14} \] ### Step 6: Simplifying the Probability However, we need to ensure that we are calculating the probability correctly. The correct approach is to consider the combinations of picking one black and one red ball without replacement: - The correct number of favorable outcomes is actually just 15 (as calculated in the first scenario) because we are not counting the same pair twice. Thus, the probability is: \[ P(\text{1 black and 1 red}) = \frac{15}{28} \] ### Conclusion: Correct Option The correct answer is **A. 15/28**. ### Explanation of Other Options - **B. 13/28**: This option is incorrect because it does not reflect the correct number of favorable outcomes. - **C. 5/14**: This option is incorrect as it simplifies the probability incorrectly. - **D. 3/14**: This option is also incorrect as it underestimates the number of favorable outcomes. ### Revision Summary - Total balls = 8 (5 black, 3 red). - Total ways to pick 2 balls = 28. - Favorable outcomes for 1 black and 1 red = 15. - Probability = 15/28, which is option A.