Question

a bag contains eight yellow marbles, nine green marbles, three purple marbles, and five red marbles. two marbles are chosen from the bag. what expression would give the probability that one marble is yellow and the other marble is red?
\\( p(y \text{ and } r) = \frac{(_{8}p_{1})(_{5}p_{1})}{_{25}p_{2}} \\)
\\( p(y \text{ and } r) = \frac{(_{8}c_{1})(_{5}c_{1})}{_{25}c_{2}} \\)
\\( p(y \text{ and } r) = \frac{(_{1}c_{8})(_{1}c_{5})}{_{2}c_{25}} \\)
\\( p(y \text{ and } r) = \frac{(_{1}p_{8})(_{1}p_{5})}{_{2}p_{25}} \\)

Explanation:

Step1: Determine Total Marbles

First, calculate the total number of marbles. Yellow (8) + Green (9) + Purple (3) + Red (5) = 25 marbles. We are choosing 2 marbles, and the order of choosing yellow and red (or red and yellow) doesn't matter here (since we just want one yellow and one red, regardless of which is picked first), so we use combinations (not permutations) for both the favorable cases and the total cases.

Step2: Favorable Cases (One Yellow, One Red)

The number of ways to choose 1 yellow marble from 8 is \( _8C_1 \), and the number of ways to choose 1 red marble from 5 is \( _5C_1 \). By the multiplication principle, the number of favorable outcomes is \( (_8C_1)(_5C_1) \).

Step3: Total Number of Ways to Choose 2 Marbles

The total number of ways to choose 2 marbles from 25 is \( _{25}C_2 \).

Step4: Probability Formula

Probability is the number of favorable outcomes divided by the total number of possible outcomes. So, \( P(Y \text{ and } R) = \frac{(_8C_1)(_5C_1)}{_{25}C_2} \).

Answer:

\( P(Y \text{ and } R) = \frac{(_8C_1)(_5C_1)}{_{25}C_2} \) (the second option)