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

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

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

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

Explanation:

Step1: Calculate total number of marbles

Total marbles \(n = 8 + 9+3 + 5=25\)

Step2: Use combination formula for probability

Probability of choosing one yellow and one red marble:
Number of ways to choose 1 yellow out of 8: \(_{8}C_{1}=\frac{8!}{1!(8 - 1)!}=8\)
Number of ways to choose 1 red out of 5: \(_{5}C_{1}=\frac{5!}{1!(5 - 1)!}=5\)
Number of ways to choose 2 marbles out of 25: \(_{25}C_{2}=\frac{25!}{2!(25 - 2)!}=\frac{25\times24}{2\times1}=300\)
Probability \(P(Y\ and\ R)=\frac{_{8}C_{1}\times_{5}C_{1}}{_{25}C_{2}}\)

Answer:

\(P(Y\ and\ R)=\frac{(_{8}C_{1})(_{5}C_{1})}{_{25}C_{2}}\) (the second option)