Kevin's Coin Problem Probability Calculation

by ADMIN 45 views
Iklan Headers

Kevin's piggy bank presents an interesting probability puzzle. Let's dive into the problem, carefully analyze the conditions, and calculate the chances of Kevin picking two coins that add up to at least 30 cents. This article will break down the problem step-by-step, ensuring a clear understanding of the solution and the underlying probability concepts.

Problem Statement: Decoding Kevin's Coin Collection

Probability is the heart of this mathematical challenge. Kevin has a collection of coins – dimes, nickels, and quarters – in his piggy bank. The key detail here is that the number of each type of coin is equal. This simplifies our calculations because it means each coin type has an equal chance of being picked. Kevin performs an interesting experiment: he randomly selects a coin, notes its value, replaces it (which is crucial information), and then selects another coin. The question is: what is the probability that the sum of the values of these two coins is at least 30 cents?

This problem involves calculating the probability of a combined event – the sum of two independent coin selections. The fact that Kevin replaces the first coin before picking the second is significant because it ensures that the two events are independent. This means the outcome of the first pick doesn't influence the outcome of the second pick. We need to consider all possible pairs of coins Kevin could pick and then determine which of these pairs add up to 30 cents or more. This requires a systematic approach to ensure we don't miss any possibilities.

Laying the Groundwork: Possible Outcomes and Coin Values

To solve this probability problem effectively, we need to first identify all the possible outcomes of Kevin's two coin selections. Since there are three types of coins (dimes, nickels, and quarters), and Kevin replaces the first coin before picking the second, there are a total of 3 * 3 = 9 possible outcomes. We can represent these outcomes as pairs, where the first coin's value is listed first, and the second coin's value is listed second. Let's use 'D' for dime (10 cents), 'N' for nickel (5 cents), and 'Q' for quarter (25 cents). The possible outcomes are:

  • (D, D)
  • (D, N)
  • (D, Q)
  • (N, D)
  • (N, N)
  • (N, Q)
  • (Q, D)
  • (Q, N)
  • (Q, Q)

Now, let's convert these coin pairs into their corresponding monetary values:

  • (10 cents, 10 cents)
  • (10 cents, 5 cents)
  • (10 cents, 25 cents)
  • (5 cents, 10 cents)
  • (5 cents, 5 cents)
  • (5 cents, 25 cents)
  • (25 cents, 10 cents)
  • (25 cents, 5 cents)
  • (25 cents, 25 cents)

This systematic listing of outcomes is a crucial step. It allows us to clearly visualize all the possibilities and ensures that we don't overlook any combinations when calculating the probability.

Calculating Probabilities: Finding Favorable Outcomes

Now that we have identified all the possible outcomes, the next crucial step is to determine which of these outcomes result in a sum of at least 30 cents. We need to go through our list of monetary values and add the two coin values in each pair. Then, we'll identify the pairs where the sum is 30 cents or more. Let's analyze each outcome:

  • (10 cents, 10 cents): 10 + 10 = 20 cents (Less than 30 cents)
  • (10 cents, 5 cents): 10 + 5 = 15 cents (Less than 30 cents)
  • (10 cents, 25 cents): 10 + 25 = 35 cents (At least 30 cents)
  • (5 cents, 10 cents): 5 + 10 = 15 cents (Less than 30 cents)
  • (5 cents, 5 cents): 5 + 5 = 10 cents (Less than 30 cents)
  • (5 cents, 25 cents): 5 + 25 = 30 cents (At least 30 cents)
  • (25 cents, 10 cents): 25 + 10 = 35 cents (At least 30 cents)
  • (25 cents, 5 cents): 25 + 5 = 30 cents (At least 30 cents)
  • (25 cents, 25 cents): 25 + 25 = 50 cents (At least 30 cents)

From this analysis, we can see that there are five favorable outcomes where the sum is at least 30 cents: (10 cents, 25 cents), (5 cents, 25 cents), (25 cents, 10 cents), (25 cents, 5 cents), and (25 cents, 25 cents). Remember, the definition of probability in this context is the number of favorable outcomes divided by the total number of possible outcomes.

Probability Formula: Putting It All Together

Now that we know the number of favorable outcomes and the total number of possible outcomes, we can calculate the probability. The formula for probability is:

Probability = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)

In this case:

  • Number of Favorable Outcomes = 5 (as we identified above)
  • Total Number of Possible Outcomes = 9 (as we calculated earlier)

Therefore, the probability that the sum of the two coins picked is at least 30 cents is:

Probability = 5 / 9

This fraction represents the likelihood of Kevin picking two coins that add up to 30 cents or more. It's a little more than 50%, indicating that it's slightly more likely than not that the sum will be at least 30 cents. Understanding this calculation provides a clear answer to the problem and demonstrates the application of basic probability principles.

Final Answer: The Probability Unveiled

In conclusion, after systematically analyzing all possible outcomes and identifying those that meet the condition of summing to at least 30 cents, we have determined that the probability is 5/9. This means that if Kevin were to repeat this coin-picking experiment many times, he would expect the sum of the two coins to be 30 cents or more in approximately 5 out of every 9 trials. This result highlights the power of probability in predicting the likelihood of events, even in seemingly simple scenarios like picking coins from a piggy bank. This problem showcases how a clear, step-by-step approach can lead to a precise and understandable answer, reinforcing the fundamental concepts of probability.

Additional insights on probability

Delving further into the concept of probability, it's essential to understand that 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. In real-world scenarios, most probabilities fall somewhere between these two extremes, representing varying degrees of likelihood.

In the context of Kevin's coin problem, we encountered what is known as classical probability. Classical probability applies when all outcomes in the sample space are equally likely, which was the case here because Kevin had an equal number of dimes, nickels, and quarters. This allowed us to simply count the favorable outcomes and divide by the total outcomes.

However, probability theory extends far beyond these simple scenarios. In many real-world situations, events are not equally likely, and we need to employ other approaches, such as empirical probability or subjective probability. Empirical probability is based on observing the frequency of an event in a series of trials. For instance, if we were to flip a coin 100 times and observe 55 heads, the empirical probability of getting heads would be 55/100 or 0.55.

Subjective probability, on the other hand, relies on personal beliefs or judgments. It's often used in situations where there is limited data or when events are unique and not easily repeatable, such as estimating the probability of a new product's success or the likelihood of a particular political outcome. These types of probability often involve expert opinions and are inherently more subjective than classical or empirical probabilities.

Understanding these different facets of probability allows us to apply probabilistic thinking to a wide array of problems, from simple coin tosses to complex financial models. Each approach offers a different lens through which to view uncertainty, enabling us to make more informed decisions in a world that is inherently probabilistic.

Expanding the problem with Conditional Probability

Another layer of complexity we can introduce to our probability discussion is the concept of conditional probability. Conditional probability deals with the probability of an event occurring given that another event has already occurred. It's a crucial concept in many real-world applications, from medical diagnosis to weather forecasting.

To illustrate conditional probability, let's modify Kevin's coin problem slightly. Suppose Kevin picks a coin, looks at it, but doesn't show us the result. He then tells us that the first coin he picked was a quarter. Now, what is the probability that the sum of the two coins he picks is at least 30 cents? This is a conditional probability problem because we are given additional information that affects the probabilities involved.

In this scenario, we are only considering outcomes where the first coin is a quarter. Our sample space has been reduced from nine possibilities to three: (Q, D), (Q, N), and (Q, Q). Among these three outcomes, two result in a sum of at least 30 cents: (Q, Q) and (Q, D) and (Q,N). Therefore, the conditional probability that the sum is at least 30 cents, given that the first coin is a quarter, is 3/3 = 1.

This example demonstrates how conditional probability can drastically change the likelihood of an event. The additional information—that the first coin was a quarter—completely shifted the probability from 5/9 to 1.

Conditional probability is formalized mathematically using the notation P(A|B), which reads as