Final answer:
To find the probability that two balls picked at random from a bag will be the same color, we need to consider different cases. We can either pick 2 red balls or 2 black balls. The probability of picking 2 red balls is 5/18 and the probability of picking 2 black balls is 1/6. Adding these probabilities together gives us an overall probability of 4/9.
Step-by-step explanation:
To find the probability that two balls picked at random from a bag will be the same color, we need to consider the different cases. We can either pick 2 red balls or 2 black balls.
Case 1: Picking 2 red balls - There are 5 red balls out of a total of 9 balls in the bag. So, the probability of picking the first red ball is 5/9. After picking the first red ball, there are 4 red balls left out of a total of 8 balls. So, the probability of picking the second red ball is 4/8.
Therefore, the probability of picking 2 red balls is (5/9) * (4/8) = 20/72 = 5/18.
Case 2: Picking 2 black balls - There are 4 black balls out of a total of 9 balls in the bag. So, the probability of picking the first black ball is 4/9. After picking the first black ball, there are 3 black balls left out of a total of 8 balls. So, the probability of picking the second black ball is 3/8.
Therefore, the probability of picking 2 black balls is (4/9) * (3/8) = 12/72 = 1/6.
Since we are considering both cases where the balls are the same color, we can add the probabilities together to get the overall probability:
P = (5/18) + (1/6) = 5/18 + 3/18 = 8/18 = 4/9
Therefore, the probability that the two balls picked at random will be the same color is 4/9.