a bag contains 10 red marbles, 15 yellow marbles, 5 green marbles, and 20 blue marbles. two marbles are…

a bag contains 10 red marbles, 15 yellow marbles, 5 green marbles, and 20 blue marbles. two marbles are drawn from the bag. which expression represents the probability that one of the marbles is red and the other is blue?\n\\(\\frac{_{30}p_{2}}{_{50}p_{2}}\\)\n\\(\\frac{_{30}c_{2}}{_{50}c_{2}}\\)\n\\(\\frac{(_{10}c_{1})(_{20}c_{1})}{_{50}c_{2}}\\)\n\\(\\frac{(_{10}p_{1})(_{20}p_{1})}{_{50}p_{2}}\\)

a bag contains 10 red marbles, 15 yellow marbles, 5 green marbles, and 20 blue marbles. two marbles are drawn from the bag. which expression represents the probability that one of the marbles is red and the other is blue?\n\\(\\frac{_{30}p_{2}}{_{50}p_{2}}\\)\n\\(\\frac{_{30}c_{2}}{_{50}c_{2}}\\)\n\\(\\frac{(_{10}c_{1})(_{20}c_{1})}{_{50}c_{2}}\\)\n\\(\\frac{(_{10}p_{1})(_{20}p_{1})}{_{50}p_{2}}\\)

Answer

Explanation:

Step1: Calculate total number of marbles

The total number of marbles is (10 + 15+5 + 20=50) marbles.

Step2: Determine the number of ways to choose 2 marbles

The number of ways to choose 2 marbles out of 50 is given by the combination formula ({n}C{r}=\frac{n!}{r!(n - r)!}), so ({50}C{2}=\frac{50!}{2!(50 - 2)!}=\frac{50\times49}{2\times 1}).

Step3: Calculate the number of ways to choose 1 red and 1 blue marble

The number of ways to choose 1 red marble out of 10 is ({10}C{1}=\frac{10!}{1!(10 - 1)!}=10), and the number of ways to choose 1 blue marble out of 20 is ({20}C{1}=\frac{20!}{1!(20 - 1)!}=20). By the multiplication - principle, the number of ways to choose 1 red and 1 blue marble is ({10}C{1}\times_{20}C_{1}=10\times20).

Step4: Calculate the probability

The probability (P) that one marble is red and the other is blue is the number of favorable outcomes (choosing 1 red and 1 blue) divided by the number of total outcomes (choosing 2 marbles out of 50). So (P = \frac{({10}C{1})({20}C{1})}{{50}C{2}}).

Answer:

(\frac{({10}C{1})({20}C{1})}{{50}C{2}})