Let's solve the problem step by step:
1. Count the total number of marbles in the bag:
[tex]\[
\text{Total number of marbles} = 8 + 9 + 3 + 5 = 25
\][/tex]
2. Calculate the number of ways to choose 1 yellow marble from 8 yellow marbles:
[tex]\[
\binom{8}{1} = 8
\][/tex]
3. Calculate the number of ways to choose 1 red marble from 5 red marbles:
[tex]\[
\binom{5}{1} = 5
\][/tex]
4. Calculate the total number of ways to choose 2 marbles out of the 25 total marbles:
[tex]\[
\binom{25}{2} = \frac{25 \times 24}{2 \times 1} = 300
\][/tex]
5. Put it all together to find the probability that one marble is yellow and the other marble is red. The expression for this probability is given by:
[tex]\[
P(Y \text{ and } R) = \frac{\binom{8}{1} \times \binom{5}{1}}{\binom{25}{2}}
\][/tex]
6. Substitute the values we calculated:
[tex]\[
P(Y \text{ and } R) = \frac{8 \times 5}{300}
\][/tex]
7. Simplify the fraction:
[tex]\[
P(Y \text{ and } R) = \frac{40}{300} = \frac{2}{15} \approx 0.1333
\][/tex]
Therefore, the probability that one marble is yellow and the other marble is red is approximately [tex]\(0.1333\)[/tex] or [tex]\(\frac{2}{15}\)[/tex]. The correct expression from the given options would correspond to:
[tex]\[
P(Y \text{ and } R) = \frac{\left(C_8 C_1\right)\left(C_5 C_1\right)}{\binom{25}{2}}
\][/tex]
where [tex]\( C_8C_1 = \binom{8}{1} \)[/tex] and [tex]\( C_5C_1 = \binom{5}{1} \)[/tex].
