Probability Of Picking Two Balls Of Different Colors

A bag contains 3 yellow and 2 red identical balls. If two balls are picked at random one after the other without replacement, what is the probability of picking two balls of different colors?

A. $\frac{3}{10}$

B. $\frac{2}{5}$

C. $\frac{3}{5}$

D. $\frac{1}{2}$

Understanding Probability

Probability, in its simplest form, quantifies the likelihood of an event occurring. It's a fundamental concept in mathematics and statistics, providing a framework for understanding and predicting outcomes in situations involving uncertainty. The probability of an event is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Events that are more likely to occur have probabilities closer to 1, while those less likely have probabilities closer to 0. To effectively grasp probability, it is essential to understand key terms such as sample space (the set of all possible outcomes), event (a subset of the sample space), and the basic formulas for calculating probabilities. For instance, the probability of an event A is often calculated as the number of favorable outcomes divided by the total number of possible outcomes. In many real-world scenarios, calculating probability involves more complex methods, such as conditional probability and the use of probability distributions. Conditional probability addresses how the probability of an event changes when another event has already occurred, while probability distributions provide a comprehensive view of all possible outcomes and their associated probabilities. Understanding these concepts allows us to analyze a wide array of situations, from simple coin flips to complex scenarios like predicting weather patterns or financial market trends. Moreover, mastering the principles of probability is crucial for making informed decisions in various fields, including science, engineering, economics, and even everyday life.

Analyzing the Problem

To tackle this specific problem, we must break down the scenario and identify the key components. We are given a bag containing 3 yellow balls and 2 red balls, making a total of 5 balls. The problem asks for the probability of picking two balls of different colors when drawing two balls one after the other without replacement. This "without replacement" condition is crucial because it means that once a ball is drawn, it is not put back into the bag, which affects the probabilities for the subsequent draw. To find the probability of picking two balls of different colors, we need to consider the possible scenarios that satisfy this condition. There are two scenarios in which we can pick balls of different colors: first picking a yellow ball and then a red ball, or first picking a red ball and then a yellow ball. We will calculate the probability of each of these scenarios separately and then add them together to find the total probability. The calculation for each scenario involves considering the probability of the first draw and then the conditional probability of the second draw, given the outcome of the first draw. For example, if we first pick a yellow ball, there will be one fewer yellow ball in the bag, and the total number of balls will also be reduced by one. This will change the probabilities for the second draw. Understanding these nuances is vital for accurately solving the problem. By carefully analyzing the problem statement and breaking it down into manageable parts, we can apply the principles of probability to arrive at the correct solution. This approach not only helps in solving this particular problem but also enhances our ability to analyze and solve similar probability-related questions in the future.

Case 1: Picking a Yellow Ball First and then a Red Ball

In the first case, let's consider the scenario where we pick a yellow ball first. Initially, there are 3 yellow balls and a total of 5 balls in the bag. The probability of picking a yellow ball on the first draw is the number of yellow balls divided by the total number of balls, which is $\frac{3}{5}$. After picking a yellow ball, there are now 2 yellow balls and 2 red balls remaining in the bag, making a total of 4 balls. To complete the first scenario, we need to pick a red ball on the second draw. Since there are 2 red balls, the probability of picking a red ball after picking a yellow ball is $\frac{2}{4}$, which simplifies to $\frac{1}{2}$. To find the probability of both events happening in sequence, we multiply the probabilities of each event. Thus, the probability of picking a yellow ball first and then a red ball is $\frac{3}{5} \times \frac{1}{2} = \frac{3}{10}$. This calculation is crucial for understanding the overall probability because it represents one of the two possible ways to pick balls of different colors. By meticulously calculating the probabilities at each step, we ensure that we accurately account for the changing composition of the bag as balls are drawn without replacement. This careful approach is vital in probability problems, where the outcome of one event can significantly impact the probabilities of subsequent events. Understanding and applying the concept of conditional probability, as demonstrated in this case, is key to mastering probability calculations.

Case 2: Picking a Red Ball First and then a Yellow Ball

Now, let's analyze the second case: picking a red ball first, followed by a yellow ball. Initially, there are 2 red balls out of a total of 5 balls in the bag. Therefore, the probability of picking a red ball on the first draw is $\frac{2}{5}$. After drawing a red ball, there are 3 yellow balls and 1 red ball remaining, making a total of 4 balls. The probability of picking a yellow ball on the second draw, given that a red ball was picked first, is the number of yellow balls divided by the new total number of balls, which is $\frac{3}{4}$. Similar to the previous case, we multiply the probabilities of each event to find the combined probability. The probability of picking a red ball first and then a yellow ball is $\frac{2}{5} \times \frac{3}{4}$. This simplifies to $\frac{6}{20}$, which further reduces to $\frac{3}{10}$. This calculation provides the probability for the second scenario where balls of different colors are picked. By carefully considering the change in the number of balls after the first draw, we ensure that the probabilities for the second draw are accurate. This step-by-step approach is essential in probability problems, as it allows us to account for the dependencies between events. Calculating the probability of each scenario separately and then combining them is a common strategy in probability, especially when dealing with mutually exclusive events. The result obtained here, $\frac{3}{10}$, represents the likelihood of this specific sequence occurring, which is a critical component in determining the overall probability of picking balls of different colors.

Calculating the Total Probability

To find the total probability of picking two balls of different colors, we combine the probabilities of the two mutually exclusive scenarios: picking a yellow ball first and then a red ball, and picking a red ball first and then a yellow ball. We calculated the probability of the first scenario as $\frac{3}{10}$ and the probability of the second scenario as $\frac{3}{10}$. Since these scenarios are mutually exclusive (they cannot both happen at the same time), we can simply add their probabilities to find the total probability. Therefore, the total probability of picking two balls of different colors is $\frac{3}{10} + \frac{3}{10}$. Adding these fractions, we get $\frac{6}{10}$, which can be simplified to $\frac{3}{5}$. This final calculation gives us the answer to the problem, representing the overall likelihood of picking two balls of different colors when drawing two balls from the bag without replacement. The process of adding probabilities for mutually exclusive events is a fundamental concept in probability theory and is widely used in various applications. By systematically breaking down the problem into scenarios, calculating the probability of each, and then combining them appropriately, we arrive at an accurate and comprehensive solution. This method not only solves the immediate problem but also enhances our understanding of probability calculations and problem-solving strategies in general.

Final Answer

The probability of picking two balls of different colors is $\frac{3}{5}$. Therefore, the correct answer is:

C. $\frac{3}{5}$