To determine the probability of drawing a yellow marble first, followed by a red marble, and finally a black marble without replacement, we need to follow these steps:
1. Calculate the total number of marbles initially:
There are 3 green, 2 yellow, 4 red, and 3 black marbles.
[tex]\[
\text{Total marbles} = 3 + 2 + 4 + 3 = 12
\][/tex]
2. Probability of drawing a yellow marble first:
The probability of picking a yellow marble first is the number of yellow marbles divided by the total number of marbles.
[tex]\[
P(\text{yellow first}) = \frac{2}{12} = \frac{1}{6} \approx 0.1667
\][/tex]
3. Adjust the total number of marbles and the number of yellow marbles after drawing the first marble:
After drawing one yellow marble, 11 marbles remain in the bag. The number of yellow marbles reduces to 1.
[tex]\[
\text{Remaining marbles} = 12 - 1 = 11
\][/tex]
4. Probability of drawing a red marble second:
The probability of picking a red marble now is the number of red marbles divided by the remaining number of marbles.
[tex]\[
P(\text{red second}) = \frac{4}{11} \approx 0.3636
\][/tex]
5. Adjust the total number of marbles and the number of red marbles after drawing the second marble:
After drawing one red marble, 10 marbles remain in the bag. The number of red marbles reduces to 3.
[tex]\[
\text{Remaining marbles} = 11 - 1 = 10
\][/tex]
6. Probability of drawing a black marble third:
The probability of picking a black marble now is the number of black marbles divided by the remaining number of marbles.
[tex]\[
P(\text{black third}) = \frac{3}{10} = 0.3
\][/tex]
7. Calculate the overall probability of these sequential events happening:
The combined probability of these independent events is the product of their individual probabilities.
[tex]\[
\text{Overall probability} = P(\text{yellow first}) \times P(\text{red second}) \times P(\text{black third})
\][/tex]
[tex]\[
= \frac{1}{6} \times \frac{4}{11} \times \frac{3}{10} = \frac{1 \cdot 4 \cdot 3}{6 \cdot 11 \cdot 10} = \frac{12}{660} = \frac{1}{55}
\][/tex]
Thus, the probability that Francisco will draw a yellow marble first, then a red marble, and then a black marble is:
[tex]\[
\boxed{\frac{1}{55}}
\][/tex]
