Probability Of Spinning One Red And One Green On A 4-Section Spinner

Hey guys! Let's dive into a probability problem that involves spinning a spinner not once, but twice! This is a classic scenario that helps us understand how probabilities work when we have multiple events. The question we're tackling today is: A spinner is spun twice, and it has 4 equal sections colored purple, red, green, and blue. What is the probability of spinning one Red and one Green? We have a few options to choose from: $\frac{1}{4}$, $\frac{1}{6}$, $\frac{1}{8}$, and $\frac{1}{16}$. Let’s break it down step by step to find the correct answer. This problem is a fantastic way to understand the fundamental principles of probability, especially when dealing with independent events. We will explore different methods to solve this, ensuring that you grasp the core concepts and can apply them to similar problems in the future. So, grab your thinking caps, and let's get started on this exciting journey of probability!

One of the most straightforward ways to solve probability problems is by listing all possible outcomes. This method is particularly useful when dealing with a limited number of events, like our spinner problem. First, let’s identify the possible outcomes for each spin. Since the spinner has 4 equally likely sections (purple, red, green, and blue), each color has an equal chance of being selected. When the spinner is spun twice, we need to consider all the combinations of outcomes from both spins. To do this systematically, we can create a table or a list. Let’s list all the possible outcomes: (Purple, Purple), (Purple, Red), (Purple, Green), (Purple, Blue), (Red, Purple), (Red, Red), (Red, Green), (Red, Blue), (Green, Purple), (Green, Red), (Green, Green), (Green, Blue), (Blue, Purple), (Blue, Red), (Blue, Green), (Blue, Blue). As you can see, there are a total of 16 possible outcomes. Now, we need to identify the outcomes where we spin one Red and one Green. These outcomes are (Red, Green) and (Green, Red). It’s crucial to include both orders since spinning Red first and then Green is different from spinning Green first and then Red. So, we have 2 favorable outcomes. The probability of spinning one Red and one Green is the number of favorable outcomes divided by the total number of possible outcomes. Therefore, the probability is $\frac{2}{16}$, which simplifies to $\frac{1}{8}$. This method provides a clear and visual way to understand the sample space and identify the outcomes that meet our criteria. By listing all possibilities, we ensure that no outcome is overlooked, leading to an accurate calculation of the probability.

Another way to approach this problem is by using the probability formula. This method involves calculating the probability of each event separately and then combining them appropriately. First, we need to determine the probability of spinning a Red section and the probability of spinning a Green section on a single spin. Since there are 4 equally likely sections, the probability of spinning Red is $\frac{1}{4}$, and the probability of spinning Green is also $\frac{1}{4}$. Now, we need to consider the two scenarios where we spin one Red and one Green: spinning Red first and then Green, or spinning Green first and then Red. The probability of spinning Red first and then Green is the product of their individual probabilities: $P(Red, Green) = P(Red) \times P(Green) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$. Similarly, the probability of spinning Green first and then Red is: $P(Green, Red) = P(Green) \times P(Red) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$. Since these are two mutually exclusive events (they can’t happen at the same time), we add their probabilities to find the total probability of spinning one Red and one Green: $P(One Red and One Green) = P(Red, Green) + P(Green, Red) = \frac{1}{16} + \frac{1}{16} = \frac{2}{16}$. Simplifying the fraction, we get $\frac{1}{8}$. This method highlights the importance of considering all possible orders of events and using the multiplication rule for independent events and the addition rule for mutually exclusive events. By breaking down the problem into smaller, manageable probabilities, we can systematically calculate the overall probability.

For those who prefer visual aids, a probability tree is an excellent tool for understanding this problem. A probability tree helps map out all possible outcomes and their respective probabilities. Let's start by drawing the first set of branches representing the first spin. There are four possibilities: Purple (P), Red (R), Green (G), and Blue (B), each with a probability of $\frac{1}{4}$. From each of these branches, we draw another set of branches representing the second spin. Again, each color (P, R, G, B) has a probability of $\frac{1}{4}$. Now, we have a tree with 16 branches, each representing a unique outcome of the two spins. To find the probability of spinning one Red and one Green, we need to identify the paths that include one Red and one Green. There are two such paths: Red then Green (R, G) and Green then Red (G, R). The probability of each path is the product of the probabilities along the path. For the path (R, G), the probability is $\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$. For the path (G, R), the probability is also $\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$. To find the total probability, we add the probabilities of these two paths: $P(One Red and One Green) = P(R, G) + P(G, R) = \frac{1}{16} + \frac{1}{16} = \frac{2}{16}$, which simplifies to $\frac{1}{8}$. The probability tree provides a visual representation of the sample space, making it easier to see all possible outcomes and calculate probabilities. It is a particularly useful tool for problems involving multiple stages or events, as it helps to organize the information and ensure that no possibilities are missed.

After exploring three different methods, we've consistently arrived at the same answer: the probability of spinning one Red and one Green is $\frac{1}{8}$. Let's delve deeper into why this is the correct answer. We’ve established that there are 16 possible outcomes when spinning the spinner twice. These outcomes are equally likely because each section of the spinner has an equal chance of being selected. The key to solving this problem is to identify the outcomes that satisfy the condition of spinning one Red and one Green. As we saw in Method 1, these outcomes are (Red, Green) and (Green, Red). It’s crucial to recognize that the order matters. Spinning Red first and then Green is a distinct outcome from spinning Green first and then Red. Therefore, we have two favorable outcomes. The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In this case, we have 2 favorable outcomes and 16 total outcomes, so the probability is $\frac{2}{16}$. Simplifying this fraction gives us $\frac{1}{8}$. Each method we used reinforces this conclusion. Listing all possible outcomes gave us a clear picture of the sample space. The probability formula allowed us to calculate the probabilities of individual events and combine them. The probability tree provided a visual representation of the process. By using multiple approaches, we’ve not only found the correct answer but also gained a deeper understanding of the underlying principles of probability. This comprehensive approach ensures that we can confidently apply these principles to other similar problems.

It's essential not only to understand why the correct answer is correct but also to understand why the other options are incorrect. This helps solidify our understanding of the problem and prevent similar mistakes in the future. Let's look at the incorrect options: $\frac{1}{4}$, $\frac{1}{6}$, and $\frac{1}{16}$.

• $\frac{1}{4}$: This option might seem plausible at first glance because there are four colors, and we want two specific colors. However, this reasoning is flawed because it doesn't account for the fact that the spinner is spun twice. The probability of spinning Red on one spin is $\frac{1}{4}$, and the probability of spinning Green on one spin is also $\frac{1}{4}$. But we need to consider the combinations and orders in which these colors can appear. Simply stating $\frac{1}{4}$ ignores the multiple possibilities that arise from two spins.
• $\frac{1}{6}$: This option is incorrect because it might arise from a misunderstanding of how to combine probabilities. There isn't a straightforward way to arrive at $\frac{1}{6}$ using the probabilities of individual spins. This option likely stems from an incorrect application of probability rules or a miscalculation of the sample space.
• $\frac{1}{16}$: This option represents the probability of spinning a specific sequence, such as Red followed by Green (or Green followed by Red). While the probability of spinning Red and then Green is indeed $\frac{1}{16}$, it only accounts for one specific order. We must remember that spinning Green and then Red is another valid outcome that satisfies the condition of spinning one Red and one Green. Therefore, $\frac{1}{16}$ only considers half of the favorable outcomes.

By understanding why these options are incorrect, we reinforce our understanding of the correct method and the nuances of probability calculations. It’s crucial to consider all possible outcomes and their respective probabilities to arrive at the correct solution.

Alright guys, we've successfully navigated this probability problem and found that the probability of spinning one Red and one Green on a spinner with four equal sections (purple, red, green, and blue) spun twice is $\frac{1}{8}$. We achieved this by using three different methods: listing all possible outcomes, applying the probability formula, and visualizing with a probability tree. Each method provided a unique perspective and reinforced our understanding of the problem. Remember, in probability problems, it’s essential to consider all possible outcomes, the order of events, and the probabilities of individual events. By breaking down the problem into smaller, manageable steps and using the appropriate formulas and techniques, we can solve even complex probability questions. We also discussed why the other options were incorrect, further solidifying our understanding of the concepts involved. Probability is a fundamental concept in mathematics and has wide-ranging applications in various fields, from statistics to finance to everyday decision-making. By mastering these basic principles, we can tackle more advanced problems and make informed decisions based on probabilities. So, keep practicing, keep exploring, and keep having fun with math!