Probability Of Selecting Coins Worth At Least 35 Cents

In this article, we will delve into a probability problem involving coin selection. The problem presents a scenario where Reyna has a collection of coins, specifically five 10-cent coins and four 25-cent coins. The core question revolves around determining the probability of Reyna selecting two coins at random that have a combined value of at least 35 cents. This problem provides an excellent opportunity to explore fundamental probability concepts and apply them in a practical context. We will break down the problem step by step, calculate the total possible outcomes, identify the favorable outcomes, and ultimately compute the desired probability. Let's embark on this journey of coin selection and probability calculation!

Understanding the Coin Selection Problem

To accurately calculate the probability, we need to first understand the different ways Reyna can select two coins from her collection. Coin selection is a critical aspect of the problem. She has a total of 9 coins (5 dimes and 4 quarters). When she chooses two coins, she's essentially creating a combination. The order in which she picks the coins doesn't matter; selecting a dime then a quarter is the same outcome as selecting a quarter then a dime for the purpose of this problem. This understanding is crucial because it dictates how we calculate the total number of possible outcomes. We'll use the concept of combinations from combinatorics to determine this, which will serve as the denominator in our probability calculation. Let's proceed to outline the detailed steps involved in solving the problem, ensuring clarity and accuracy in our calculations.

Determining the Total Number of Outcomes

To solve this probability problem, the first critical step is to determine the total number of possible outcomes. In this case, an outcome represents a unique pair of coins that Reyna can select from her collection of nine coins (five 10-cent coins and four 25-cent coins). Since the order in which the coins are selected does not matter, we are dealing with combinations rather than permutations. The formula for calculating combinations is denoted as nCr, which is expressed as n! / (r! * (n-r)!), where n is the total number of items, and r is the number of items being chosen. In our scenario, n equals 9 (the total number of coins), and r equals 2 (the number of coins Reyna selects). Applying this formula will give us the total number of ways Reyna can choose two coins from her collection. This number will be the denominator in our probability calculation, representing the total possible outcomes. Accurately calculating this value is essential for determining the probability of selecting coins worth at least 35 cents.

Calculating the total number of ways to choose 2 coins from 9 involves using the combinations formula. The formula for combinations is nCr = n! / (r! * (n-r)!), where n is the total number of items, r is the number of items to choose, and "!" denotes the factorial. In this case, n = 9 (total number of coins) and r = 2 (number of coins to choose). Plugging these values into the formula, we get 9C2 = 9! / (2! * (9-2)!) = 9! / (2! * 7!). Expanding the factorials, we have 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1, 2! = 2 × 1 = 2, and 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1. To simplify the calculation, we can write 9! as 9 × 8 × 7! and then cancel out the 7! in the numerator and denominator. This leaves us with 9C2 = (9 × 8) / 2. Further simplifying, we get 9C2 = 72 / 2 = 36. Therefore, there are 36 different ways for Reyna to choose two coins from her collection of nine coins. This value represents the total number of possible outcomes when selecting two coins, which is a crucial component in calculating the probability of the desired event.

Identifying Favorable Outcomes

After determining the total number of possible outcomes, the next crucial step is to identify the favorable outcomes. In this context, favorable outcomes are the combinations of two coins that have a combined value of at least 35 cents. Reyna has two types of coins: 10-cent coins (dimes) and 25-cent coins (quarters). To achieve a total value of at least 35 cents, there are a few possible combinations. One such combination is selecting two quarters, which would give a total of 50 cents. Another way to reach at least 35 cents is by selecting one dime and one quarter, resulting in a total of 35 cents. It's not possible to reach 35 cents or more by selecting only two dimes, as their combined value would only be 20 cents. Therefore, we need to count the number of ways Reyna can select two quarters and the number of ways she can select one dime and one quarter. Summing these counts will give us the total number of favorable outcomes, which we will then use to calculate the probability. Identifying and accurately counting these favorable outcomes is essential for solving the problem correctly.

To enumerate the favorable outcomes, let's consider each possible combination of coins that results in a total value of at least 35 cents. As established earlier, these combinations are two quarters and one dime combined with one quarter. First, let's calculate the number of ways to choose two quarters. Reyna has four quarters, and she needs to choose two. Using the combinations formula, we calculate 4C2 = 4! / (2! * (4-2)!) = 4! / (2! * 2!) = (4 × 3 × 2 × 1) / ((2 × 1) × (2 × 1)) = (4 × 3) / (2 × 1) = 12 / 2 = 6. So, there are 6 ways to choose two quarters. Next, let's calculate the number of ways to choose one dime and one quarter. Reyna has five dimes and four quarters. The number of ways to choose one dime is 5C1 = 5, and the number of ways to choose one quarter is 4C1 = 4. To find the total number of ways to choose one dime and one quarter, we multiply these two values together: 5 × 4 = 20. Therefore, there are 20 ways to choose one dime and one quarter. Now, we add the number of ways to choose two quarters and the number of ways to choose one dime and one quarter to get the total number of favorable outcomes: 6 + 20 = 26. Thus, there are 26 favorable outcomes where the combined value of the two selected coins is at least 35 cents. This total is crucial for the final step of calculating the probability.

Calculating the Probability

With the total number of possible outcomes and the number of favorable outcomes determined, we can now calculate the probability of Reyna selecting two coins worth at least 35 cents. Probability calculation is the final step. Probability is defined as the ratio of favorable outcomes to total possible outcomes. In this scenario, the number of favorable outcomes is 26, as we calculated there are 26 ways for Reyna to select two coins with a combined value of at least 35 cents. The total number of possible outcomes is 36, which represents all possible pairs of coins Reyna can select from her collection. Therefore, the probability is calculated as 26 divided by 36. This fraction can then be simplified to its lowest terms to express the probability in its simplest form. The resulting fraction or decimal will give us the likelihood of Reyna selecting two coins that meet the specified value threshold. Let's proceed with the calculation to arrive at the final answer.

To calculate the probability, we divide the number of favorable outcomes by the total number of possible outcomes. We have determined that there are 26 favorable outcomes (combinations of coins worth at least 35 cents) and 36 total possible outcomes (all possible combinations of two coins). Therefore, the probability is 26/36. To simplify this fraction, we look for the greatest common divisor (GCD) of 26 and 36. The GCD of 26 and 36 is 2. Dividing both the numerator and the denominator by 2, we get 26 ÷ 2 = 13 and 36 ÷ 2 = 18. Thus, the simplified fraction is 13/18. This means that the probability of Reyna selecting two coins with a combined value of at least 35 cents is 13 out of 18. This probability can also be expressed as a decimal by dividing 13 by 18, which gives approximately 0.7222. Therefore, there is a 13/18 or approximately 72.22% chance that the two coins Reyna selects will be worth at least 35 cents. This completes our calculation of the probability.

Conclusion

In conclusion, we have successfully calculated the probability of Reyna selecting two coins worth at least 35 cents from her collection of five 10-cent coins and four 25-cent coins. By systematically breaking down the problem, we first determined the total number of possible outcomes, which was found to be 36, using the combinations formula. Next, we identified the favorable outcomes, which are the combinations of coins with a combined value of at least 35 cents. We found that there were 26 such combinations. Finally, we calculated the probability by dividing the number of favorable outcomes by the total number of possible outcomes, resulting in a probability of 26/36, which simplifies to 13/18 or approximately 72.22%. This problem illustrates the application of fundamental probability concepts in a real-world scenario and highlights the importance of carefully considering all possible outcomes and favorable outcomes in probability calculations. The solution demonstrates a clear and methodical approach to solving probability problems, which can be applied to various other situations involving chance and selection.