Probability Of Drawing A Black Sock Then A Red Sock

In the realm of probability, understanding how to calculate the likelihood of sequential events is crucial. Probability problems involving dependent and independent events often appear in various contexts, from games of chance to real-world scenarios. This article delves into a specific probability question: Luka has a bag containing 5 socks: 3 red, 1 white, and 1 black. He draws 1 sock out of the bag, replaces it, and draws another sock. What is the probability that he will draw a black sock and then a red sock? This seemingly simple problem opens the door to exploring fundamental probability concepts, including independent events, conditional probability, and calculating combined probabilities.

In this comprehensive analysis, we will first break down the problem statement to ensure a clear understanding of the given conditions. We will then discuss the principles of probability that apply to this scenario, specifically focusing on how replacement affects the probabilities of subsequent draws. Following this, we will calculate the probability of drawing a black sock on the first draw and the probability of drawing a red sock on the second draw, taking into account the replacement. Finally, we will combine these probabilities to determine the overall probability of the specific sequence of events: drawing a black sock followed by a red sock. By the end of this discussion, you will not only understand the solution to this particular problem but also gain a deeper insight into probability calculations involving multiple events.

Before diving into the calculations, it is essential to dissect the problem statement thoroughly. Luka's bag contains a total of 5 socks, comprising 3 red socks, 1 white sock, and 1 black sock. The key here is to recognize the distribution of socks by color, as this will directly influence the probabilities of drawing each color. The problem introduces a two-step process: Luka draws a sock, notes its color, replaces it back into the bag, and then draws another sock. The replacement aspect is crucial because it ensures that the probabilities for the second draw remain the same as the first draw. This is because the total number of socks and the composition of colors in the bag are reset after each draw.

The question asks for the probability of a specific sequence of events: first, drawing a black sock, and second, drawing a red sock. To solve this, we need to consider each draw as an independent event due to the replacement. This means the outcome of the first draw does not affect the probabilities in the second draw. We will calculate the probability of each event separately and then combine them to find the probability of the combined event. Understanding this independence is vital for correctly applying the multiplication rule of probability, which we will use later in the solution. The problem's clarity in specifying the order of draws (black sock first, then red sock) is also important, as changing the order would change the probabilities being calculated.

To effectively tackle the sock-drawing problem, it's essential to establish a solid foundation in the basic principles of probability. At its core, probability is the 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. The fundamental formula for calculating the probability of an event is:

Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes)

In the context of our problem, a favorable outcome is the specific event we are interested in (e.g., drawing a black sock), and the total number of possible outcomes is the total number of socks in the bag. This basic formula is the building block for more complex probability calculations. When dealing with multiple events, we need to consider whether these events are independent or dependent. Independent events are events where the outcome of one does not affect the outcome of the other. In Luka's sock problem, the act of replacing the sock after the first draw makes the two draws independent events. This is a crucial distinction because the probability of the second event remains constant regardless of the outcome of the first. Conversely, dependent events are those where the outcome of one event does influence the outcome of the subsequent event. Understanding the distinction between independent and dependent events is vital for selecting the correct approach to probability calculations. For independent events, we can use the multiplication rule, which we will explore in more detail in the next section.

In the sock-drawing scenario, the concept of independent events is pivotal. Events are considered independent when the outcome of one event does not impact the outcome of another. In Luka's case, because he replaces the sock after each draw, the composition of the bag remains constant for the second draw, thereby making the two draws independent events. This independence significantly simplifies the probability calculation.

When dealing with independent events, the multiplication rule is the key to finding the probability of both events occurring in sequence. The multiplication rule states that the probability of two independent events A and B both occurring is the product of their individual probabilities. Mathematically, this is represented as:

P(A and B) = P(A) * P(B)

Where P(A) is the probability of event A occurring, and P(B) is the probability of event B occurring. Applying this to Luka's sock problem, we can define event A as drawing a black sock on the first draw and event B as drawing a red sock on the second draw. To find the probability of Luka drawing a black sock and then a red sock, we need to calculate P(black sock on the first draw) and P(red sock on the second draw) separately and then multiply these probabilities together. The replacement of the sock ensures that the total number of socks and the number of socks of each color remain the same for both draws, maintaining the independence of the events. This application of the multiplication rule is fundamental to solving the problem accurately.

To calculate the probability of Luka drawing a black sock on the first draw, we need to apply the basic probability formula: Probability = (Number of favorable outcomes) / (Total number of possible outcomes). In this case, the favorable outcome is drawing a black sock. Looking at the contents of the bag, there is 1 black sock. The total number of possible outcomes is the total number of socks in the bag, which is 5 (3 red, 1 white, and 1 black). Therefore, the probability of drawing a black sock on the first draw is:

P(Black Sock First) = (Number of Black Socks) / (Total Number of Socks) = 1 / 5

This fraction, 1/5, represents the likelihood of Luka drawing a black sock in his first attempt. We can also express this probability as a decimal (0.2) or a percentage (20%). This means that there is a 20% chance that Luka will draw a black sock on his first draw. Understanding this individual probability is a critical step in solving the overall problem, as we will later multiply this probability with the probability of drawing a red sock on the second draw. The straightforward application of the probability formula here highlights the importance of accurately identifying both the favorable outcomes and the total possible outcomes in a given scenario. This calculation forms the first half of our solution, setting the stage for determining the probability of the second event.

Following the determination of the probability of drawing a black sock first, the next crucial step is to calculate the probability of Luka drawing a red sock on the second draw. Remember, Luka replaces the sock he draws on the first attempt, ensuring that the conditions for the second draw are identical to those of the first. This means the total number of socks in the bag remains at 5, and the number of red socks remains at 3.

Using the basic probability formula again, Probability = (Number of favorable outcomes) / (Total number of possible outcomes), we can calculate the probability of drawing a red sock. Here, the favorable outcome is drawing a red sock, and there are 3 red socks in the bag. The total number of possible outcomes is still 5, the total number of socks. Therefore, the probability of drawing a red sock on the second draw is:

P(Red Sock Second) = (Number of Red Socks) / (Total Number of Socks) = 3 / 5

This fraction, 3/5, represents the likelihood of Luka drawing a red sock on his second attempt. Expressed as a decimal, this probability is 0.6, or 60% as a percentage. This indicates a higher chance of drawing a red sock compared to the black sock due to the greater number of red socks in the bag. With both individual probabilities now calculated – the probability of drawing a black sock first and the probability of drawing a red sock second – we are ready to combine these probabilities to find the overall probability of the sequence of events. The next step involves applying the multiplication rule for independent events, which will give us the final answer to the problem.

Having calculated the individual probabilities of Luka drawing a black sock first (1/5) and a red sock second (3/5), we now combine these probabilities to find the overall probability of both events occurring in the specified sequence. Since the events are independent—Luka replaces the first sock before drawing the second—we use the multiplication rule. This rule states that the probability of two independent events A and B both occurring is the product of their individual probabilities:

P(A and B) = P(A) * P(B)

In this scenario, event A is drawing a black sock on the first draw, and event B is drawing a red sock on the second draw. Therefore, the overall probability of Luka drawing a black sock first and then a red sock is:

P(Black Sock First and Red Sock Second) = P(Black Sock First) * P(Red Sock Second)

Substituting the values we calculated earlier:

P(Black Sock First and Red Sock Second) = (1/5) * (3/5)

Multiplying these fractions gives us:

P(Black Sock First and Red Sock Second) = 3 / 25

Thus, the probability that Luka will draw a black sock first and then a red sock is 3/25. This fraction can also be expressed as a decimal (0.12) or a percentage (12%). This means there is a 12% chance that Luka will draw a black sock followed by a red sock in this scenario. This final calculation provides the answer to the original problem, demonstrating the power of understanding and applying basic probability principles to solve compound event scenarios.

In conclusion, the problem of determining the probability of Luka drawing a black sock followed by a red sock from a bag of 5 socks (3 red, 1 white, and 1 black), with replacement, illustrates several fundamental concepts in probability. We began by dissecting the problem statement, emphasizing the importance of understanding the given conditions, particularly the act of replacement, which establishes the independence of the two draws. We then reviewed basic probability principles, including the definition of probability and the distinction between independent and dependent events. Central to our solution was the application of the multiplication rule for independent events, which allowed us to calculate the overall probability by multiplying the individual probabilities of each event.

We methodically calculated the probability of drawing a black sock first (1/5) and the probability of drawing a red sock second (3/5). Finally, we combined these probabilities using the multiplication rule to arrive at the overall probability of 3/25, or 12%. This result demonstrates that while there is a possibility of Luka drawing a black sock followed by a red sock, the probability is relatively low due to the limited number of black socks in the bag. This exercise not only provides a specific answer to the posed question but also reinforces a broader understanding of how to approach and solve probability problems involving sequential events. By breaking down the problem into manageable steps and applying the appropriate principles, we can confidently navigate similar scenarios in probability and statistics.