Question

5
select the correct answer.
kevin has an equal number of dimes, nickels, and quarters in his piggy bank. he randomly picks a coin, replaces it, and picks another coin. what is the probability that the sum of the two coins picked is at least 30 cents?
a. \\(\frac{2}{9}\\)
b. \\(\frac{1}{3}\\)
c. \\(\frac{4}{9}\\)
d. \\(\frac{5}{9}\\)

Explanation:

Step1: Define the coins and their values

Let \( N \) represent nickel (5 cents), \( D \) represent dime (10 cents), and \( Q \) represent quarter (25 cents). Since there are equal numbers of each, the probability of picking each coin is \( \frac{1}{3} \) for each pick (with replacement, so independent events).

Step2: List all possible pairs of coin picks

There are \( 3\times3 = 9 \) possible ordered pairs: \( (N,N) \), \( (N,D) \), \( (N,Q) \), \( (D,N) \), \( (D,D) \), \( (D,Q) \), \( (Q,N) \), \( (Q,D) \), \( (Q,Q) \).

Step3: Calculate the sum for each pair

• \( (N,N) \): \( 5 + 5 = 10 \) cents
• \( (N,D) \): \( 5 + 10 = 15 \) cents
• \( (N,Q) \): \( 5 + 25 = 30 \) cents
• \( (D,N) \): \( 10 + 5 = 15 \) cents
• \( (D,D) \): \( 10 + 10 = 20 \) cents
• \( (D,Q) \): \( 10 + 25 = 35 \) cents
• \( (Q,N) \): \( 25 + 5 = 30 \) cents
• \( (Q,D) \): \( 25 + 10 = 35 \) cents
• \( (Q,Q) \): \( 25 + 25 = 50 \) cents

Step4: Identify pairs with sum at least 30 cents

The pairs are \( (N,Q) \), \( (D,Q) \), \( (Q,N) \), \( (Q,D) \), \( (Q,Q) \). That's 5 pairs.

Step5: Calculate the probability

The probability is the number of favorable outcomes divided by total outcomes. So \( P = \frac{5}{9} \).

Answer:

D. \(\frac{5}{9}\)