Final answer:
The probability that Karen pulls out a red marble first and then a blue marble, without replacement, is 1/11.
Step-by-step explanation:
To calculate the probability that the first marble Karen pulls out is red, and the second one is blue, we take the following steps:
- Determine the total number of marbles: 4 red + 5 green + 3 blue = 12 marbles.
- Calculate the probability of drawing a red marble first: There are 4 red marbles out of 12, so P(first is red) = 4/12 or 1/3.
- After drawing a red marble, there are now 11 marbles left: 3 red, 5 green, and 3 blue.
- Calculate the probability of drawing a blue marble second: Now, since one red marble is removed, there are 3 blue marbles out of 11, so P(second is blue after red) = 3/11.
- To get the overall probability of both events happening in sequence, we multiply the individual probabilities: P(first is red and second is blue) = P(first is red) × P(second is blue after red) = (1/3) × (3/11).
- The result is 3/33 or 1/11.
Therefore, the probability that the first marble is red and the second is blue is 1/11.