Question

there are 10 cups placed facedown on a table. beneath each cup is a coin. together, there are 5 quarters, 2 nickels, and 3 dimes. one cup is selected at random, replaced, and then another cup is selected. what is the probability of selecting a quarter followed by a nickel? 7/10 3/20 1/9 1/10

Explanation:

Step1: Calculate the probability of selecting a quarter

The probability of selecting a quarter is the number of quarters divided by the total number of cups. There are 5 quarters and 10 cups in total. So the probability \(P(\text{quarter})=\frac{5}{10}=\frac{1}{2}\)

Step2: Calculate the probability of selecting a nickel

Since the cup is replaced, the total number of cups remains 10. There are 2 nickels. So the probability \(P(\text{nickel})=\frac{2}{10}=\frac{1}{5}\)

Step3: Calculate the probability of both events

Since the two events (selecting a quarter and then a nickel) are independent (because of replacement), we use the multiplication rule for independent events \(P(A\cap B)=P(A)\times P(B)\). Here \(A\) is selecting a quarter and \(B\) is selecting a nickel. So \(P = \frac{1}{2}\times\frac{1}{5}=\frac{1}{10}\)

Answer:

\(\frac{1}{10}\)