Final answer:
The probability of picking a black marble, putting it back into the bag, and then picking another black marble is 4/9. This is calculated by multiplying the individual probabilities of picking a black marble for each draw, with the result (2/3) × (2/3) = 4/9. The provided options do not include the correct answer.
Step-by-step explanation:
The question asks about the probability of picking a black marble, putting it back, and then picking another black marble from a bag with a certain composition of marbles. To solve this, we consider each event independently because the marble is put back after each draw, maintaining the initial conditions of the bag for each draw.
To find the probability of picking a black marble first, we use the fact that there are two black marbles and one red marble in the bag. So, the probability P(Black first) = Number of black marbles / Total number of marbles = 2/3.
Since the black marble is put back after the first draw, the conditions of the bag remain unchanged. Therefore, the probability of picking a black marble again is the same as the first time, which is 2/3. To find the combined probability of both events happening one after the other (independent events), we multiply their probabilities: P(Black first) × P(Black second) = (2/3) × (2/3).
After calculating the multiplication, we get 4/9 as the probability of picking a black marble first and then picking a black marble again after replacing the first one back in the bag. Hence, none of the provided options A. 1/6, B. 1/2, C. 2/3, D. 3/4, are correct. The correct answer should be (4/9), which is not listed in the provided options.