Probability Of Summing Coins To At Least 30 Cents

This article dives into a probability problem involving coin selection. The core challenge is to determine the likelihood of drawing two coins (with replacement) from a collection of dimes, nickels, and quarters such that their combined value is 30 cents or more. This problem combines basic probability principles with an understanding of coin values, providing an engaging exercise in mathematical reasoning. We'll explore the problem setup, methodically calculate the probabilities of favorable outcomes, and arrive at the final solution.

Kevin's piggy bank contains an equal number of dimes (10 cents), nickels (5 cents), and quarters (25 cents). Kevin randomly selects a coin, notes its value, replaces it, and then selects another coin. The question we aim to answer is: What is the probability that the sum of the values of the two coins Kevin picks is at least 30 cents?

Before diving into the specifics of this coin problem, let's revisit the fundamental concepts of probability. Probability is a measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The probability of an event is often expressed as a fraction, decimal, or percentage.

When dealing with multiple events, such as drawing two coins in this case, we need to consider the sample space, which is the set of all possible outcomes. Since Kevin replaces the coin after each draw, the two events are independent. This means the outcome of the first draw does not affect the outcome of the second draw. The probability of two independent events both occurring is the product of their individual probabilities.

In this problem, we will calculate the probabilities of the coin draws that result in a sum of 30 cents or more. This involves identifying the favorable outcomes, determining the probability of each outcome, and then summing these probabilities to find the overall probability.

The first step in solving this problem is to define the possible outcomes when Kevin picks two coins. Since there are three types of coins (dimes, nickels, and quarters), there are 3 * 3 = 9 possible pairs of coins that Kevin could pick. These outcomes, considering the order in which they are picked, are:

1. Dime, Dime (10 cents + 10 cents = 20 cents)
2. Dime, Nickel (10 cents + 5 cents = 15 cents)
3. Dime, Quarter (10 cents + 25 cents = 35 cents)
4. Nickel, Dime (5 cents + 10 cents = 15 cents)
5. Nickel, Nickel (5 cents + 5 cents = 10 cents)
6. Nickel, Quarter (5 cents + 25 cents = 30 cents)
7. Quarter, Dime (25 cents + 10 cents = 35 cents)
8. Quarter, Nickel (25 cents + 5 cents = 30 cents)
9. Quarter, Quarter (25 cents + 25 cents = 50 cents)

Each of these outcomes is equally likely because Kevin has an equal number of each type of coin and replaces the coin after each draw. Now, we can identify the outcomes where the sum is at least 30 cents.

Now, let's pinpoint the outcomes where the sum of the two coins is 30 cents or more. Looking at the list above, we can identify the following favorable outcomes:

• Dime, Quarter (10 cents + 25 cents = 35 cents)
• Nickel, Quarter (5 cents + 25 cents = 30 cents)
• Quarter, Dime (25 cents + 10 cents = 35 cents)
• Quarter, Nickel (25 cents + 5 cents = 30 cents)
• Quarter, Quarter (25 cents + 25 cents = 50 cents)

There are 5 favorable outcomes out of the 9 total possible outcomes. This information is crucial for calculating the probability of the event we're interested in. The next step involves calculating the probability of each individual outcome and then combining these probabilities.

Since Kevin has an equal number of dimes, nickels, and quarters, the probability of picking any one type of coin on a single draw is 1/3. This is because there are three equally likely possibilities. Because Kevin replaces the coin each time, the draws are independent events. The probability of two independent events occurring is the product of their individual probabilities.

Let's consider one of the favorable outcomes: picking a dime first and then a quarter. The probability of picking a dime is 1/3, and the probability of picking a quarter is also 1/3. Therefore, the probability of picking a dime and then a quarter is (1/3) * (1/3) = 1/9. This same probability applies to each of the other possible combinations of coins.

Now that we've calculated the probability of each individual outcome (1/9), and we know there are 5 favorable outcomes, we can calculate the overall probability of picking two coins with a sum of at least 30 cents. This is simply the sum of the probabilities of each favorable outcome. Since each outcome has a probability of 1/9, and there are 5 favorable outcomes, the total probability is:

5 * (1/9) = 5/9

Therefore, the probability that the sum of the two coins Kevin picks is at least 30 cents is 5/9. This result represents the likelihood of the combined value of the selected coins meeting or exceeding the specified threshold.

Another way to visualize and solve this problem is by constructing a probability table. This table will show all possible outcomes of the two coin selections and their corresponding sums. The table helps in organizing the information and easily identifying the favorable outcomes.

In this table, each cell represents a possible outcome, and the value in the cell is the sum of the two coins. There are 9 total outcomes, as we previously determined. By counting the outcomes with a sum of 30 cents or more, we can easily see that there are 5 favorable outcomes.

The probability of each outcome remains 1/9, as before. The overall probability of the sum being at least 30 cents is still calculated by summing the probabilities of the favorable outcomes, resulting in 5/9.

In conclusion, the probability that the sum of the two coins Kevin picks is at least 30 cents is 5/9. We arrived at this solution by first defining the sample space of all possible outcomes, then identifying the favorable outcomes (those with a sum of 30 cents or more), and finally calculating the probability of each outcome and summing them up. We also explored an alternative approach using a probability table to visualize and solve the problem. This problem illustrates the fundamental principles of probability and how they can be applied to practical scenarios involving multiple events.

Probability, Coin Selection, Dime, Nickel, Quarter, Independent Events, Sample Space, Favorable Outcomes, Probability Calculation, Probability Table, Combinations, Sum of Coins, Mathematical Reasoning, Likelihood, Coin Values