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?

$p(y\text{ and }r)=\frac{(_{8}p_{1})(_{5}p_{1})}{_{25}p_{2}}$

$p(y\text{ and }r)=\frac{(_{8}c_{1})(_{5}c_{1})}{_{25}c_{2}}$

$p(y\text{ and }r)=\frac{(_{5}c_{8})(_{5}c_{5})}{_{5}c_{25}}$

$p(y\text{ and }r)=\frac{(_{1}p_{8})(_{1}p_{5})}{_{2}p_{25}}$

Explanation:

Step1: Calculate total number of marbles

Total marbles = 8 (yellow) + 9 (green) + 3 (purple) + 5 (red) = 25 marbles.

Step2: Use combination formula for probability

The probability of choosing one yellow marble out of 8 and one red marble out of 5 when choosing 2 marbles out of 25 is given by the formula for combinations. The number of ways to choose 1 yellow marble out of 8 is $_{8}C_{1}$, the number of ways to choose 1 red marble out of 5 is $_{5}C_{1}$, and the number of ways to choose 2 marbles out of 25 is $_{25}C_{2}$. So the probability $P(Y\text{ and }R)=\frac{_{8}C_{1}\times_{5}C_{1}}{_{25}C_{2}}$.

Answer:

$P(Y\text{ and }R)=\frac{_{8}C_{1}\times_{5}C_{1}}{_{25}C_{2}}$ (corresponding to the second option in the multiple - choice list you provided)