To determine the probability that one marble is yellow and the other marble is red when two marbles are chosen from a bag containing eight yellow marbles, nine green marbles, three purple marbles, and five red marbles, we will follow these steps:
1. Calculate the total number of marbles:
[tex]\[
\text{Total marbles} = 8 \text{ (yellow)} + 9 \text{ (green)} + 3 \text{ (purple)} + 5 \text{ (red)} = 25
\][/tex]
2. Determine the number of ways to choose 2 marbles out of the 25:
This is a combination problem where we need to choose 2 marbles out of 25 without regard to order. We use the combination formula [tex]\( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)[/tex]:
[tex]\[
\binom{25}{2} = \frac{25!}{2!(25-2)!} = \frac{25 \times 24}{2 \times 1} = 300
\][/tex]
3. Determine the number of ways to choose 1 yellow marble and 1 red marble:
- The number of ways to choose 1 yellow marble from 8 is [tex]\( \binom{8}{1} \)[/tex]:
[tex]\[
\binom{8}{1} = 8
\][/tex]
- The number of ways to choose 1 red marble from 5 is [tex]\( \binom{5}{1} \)[/tex]:
[tex]\[
\binom{5}{1} = 5
\][/tex]
- The number of ways to choose 1 yellow marble and 1 red marble is the product of these two combinations:
[tex]\[
8 \times 5 = 40
\][/tex]
4. Calculate the probability:
The probability is the number of favorable outcomes (choosing 1 yellow and 1 red marble) divided by the total number of possible outcomes (choosing 2 marbles out of 25):
[tex]\[
P(\text{Y and R}) = \frac{\binom{8}{1} \cdot \binom{5}{1}}{\binom{25}{2}} = \frac{40}{300} = \frac{2}{15}
\][/tex]
Hence, the correct expression that gives the probability that one marble is yellow and the other marble is red is:
[tex]\[
P(\text{Y and R}) = \frac{\left({ }_8 C_1\right)\left({ }_5 C_1\right)}{{ }_{25} C_2}
\][/tex]
The numerical result from this calculation confirms the probability is approximately [tex]\(0.1333\)[/tex] or [tex]\(\frac{2}{15}\)[/tex].
