Sure, let's break this problem down step-by-step.
### Step 1: Determine the Total Number of Marbles
First, we need to understand the composition of the marbles in the bag:
- Red marbles: 3
- Yellow marbles: 2
- Green marbles: 3
So, the total number of marbles in the bag is:
[tex]\[ 3 + 2 + 3 = 8 \][/tex]
### Step 2: Probability of Drawing the First Green Marble
To find the probability of drawing a green marble first, we look at the number of green marbles and the total marbles:
[tex]\[ \text{Probability of drawing first green marble} = \frac{\text{Number of green marbles}}{\text{Total number of marbles}} = \frac{3}{8} = 0.375 \][/tex]
### Step 3: Probability of Drawing the Second Green Marble
After drawing one green marble, there are now 2 green marbles left and a total of 7 marbles remaining in the bag.
[tex]\[ \text{Probability of drawing second green marble} = \frac{\text{Remaining green marbles}}{\text{Remaining total marbles}} = \frac{2}{7} \approx 0.2857 \][/tex]
### Step 4: Probability of Drawing a Yellow Marble on the Third Draw
After drawing two green marbles, there are 2 yellow marbles left and a total of 6 marbles remaining in the bag.
[tex]\[ \text{Probability of drawing third yellow marble} = \frac{\text{Remaining yellow marbles}}{\text{Remaining total marbles}} = \frac{2}{6} = \frac{1}{3} \approx 0.3333 \][/tex]
### Step 5: Combining the Probabilities
The events are sequential and without replacement, so the overall probability is the product of the individual probabilities:
[tex]\[ \text{Overall Probability} = \left( \frac{3}{8} \right) \times \left( \frac{2}{7} \right) \times \left( \frac{1}{3} \right) \][/tex]
### Final Computation
[tex]\[ \text{Overall Probability} = 0.375 \times 0.2857 \times 0.3333 \approx 0.0357 \][/tex]
Hence, the probability that the first two marbles drawn are green and the third marble drawn is yellow is approximately:
[tex]\[ 0.0357 \][/tex]
So there you have it! This is a step-by-step solution showing how to find the probability that the first two marbles drawn are green and the third is yellow, which is approximately [tex]\(0.0357\)[/tex].
