A bag contains 10 red marbles, five blue marbles, and nine yellow

marbles. Marbles are taken from the bag without replacement until 3 marbles are drawn. Find the probability distribution for choosing yellow marbles using combinations.

To find the probability distribution for choosing yellow marbles, we need to consider all possible combinations of drawing 3 marbles without replacement from the bag.

First, let's calculate the total number of combinations of drawing 3 marbles from a bag of 24 marbles:

Total combinations = C(24, 3) = (24! / (3! * (24-3)!)) = 2024

Now, let's calculate the number of combinations that result in selecting a certain number of yellow marbles:

For selecting 0 yellow marbles:
The number of ways to choose 3 marbles without any yellow marbles = C(15, 3) = (15! / (3! * (15-3)!)) = 455

For selecting 1 yellow marble:
The number of ways to choose 2 non-yellow marbles and 1 yellow marble = C(15, 1) * C(9, 2) = (15! / (1! * (15-1)!)) * (9! / (2! * (9-2)!)) = 15 * 36 = 540

For selecting 2 yellow marbles:
The number of ways to choose 1 non-yellow marble and 2 yellow marbles = C(15, 2) * C(9, 1) = (15! / (2! * (15-2)!)) * (9! / (1! * (9-1)!)) = 105 * 9 = 945

For selecting all 3 yellow marbles:
The number of ways to choose 3 yellow marbles = C(9, 3) = (9! / (3! * (9-3)!)) = 84

Now, let's calculate the probabilities for each event by dividing the number of combinations by the total combinations:

Probability of selecting 0 yellow marbles = 455 / 2024 ≈ 0.225
Probability of selecting 1 yellow marble = 540 / 2024 ≈ 0.267
Probability of selecting 2 yellow marbles = 945 / 2024 ≈ 0.467
Probability of selecting 3 yellow marbles = 84 / 2024 ≈ 0.041

Therefore, the probability distribution for choosing yellow marbles using combinations is approximately as follows:
P(X = 0) ≈ 0.225
P(X = 1) ≈ 0.267
P(X = 2) ≈ 0.467
P(X = 3) ≈ 0.041