Probability Of Drawing Socks A Step-by-Step Solution

In the realm of probability, understanding how to calculate the likelihood of sequential events is a fundamental skill. This article delves into a classic probability problem involving drawing socks from a bag, focusing on the probability of drawing a black sock followed by a red sock. We'll break down the problem step-by-step, ensuring a clear understanding of the concepts involved. The specific problem we'll address is: 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 then draws another sock. What is the probability that he will draw a black sock and then a red sock, P(black, then red)? This problem exemplifies the concept of independent events, where the outcome of the first event doesn't affect the outcome of the second event due to the replacement of the sock. Understanding this principle is crucial for tackling various probability problems.

Understanding the Basics of Probability

To solve this problem effectively, it's essential to grasp the basic principles of probability. 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. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In mathematical terms:

P(Event) = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)

For instance, if we consider a fair six-sided die, the probability of rolling a 4 is 1/6 because there is only one face with a 4, and there are six possible outcomes in total. Similarly, the probability of rolling an even number (2, 4, or 6) is 3/6, which simplifies to 1/2. This fundamental concept forms the bedrock for understanding more complex probability scenarios, such as the sock-drawing problem we're about to explore.

In scenarios involving multiple events, such as drawing socks in sequence, we often deal with the concept of independent events and dependent events. Independent events are those where the outcome of one event does not influence the outcome of another. For example, flipping a coin multiple times are independent events, as the result of one flip does not change the probability of the next flip. Dependent events, on the other hand, are those where the outcome of one event affects the outcome of another. Drawing cards from a deck without replacement is a classic example of dependent events, as removing a card changes the composition of the remaining deck and thus the probabilities of subsequent draws. The sock-drawing problem in question involves independent events because Luka replaces the sock after the first draw, restoring the original composition of the bag.

Step-by-Step Solution

Now, let's dissect the sock-drawing problem step by step. The problem states that Luka has a bag with 5 socks: 3 red, 1 white, and 1 black. He draws a sock, replaces it, and then draws another. We need to find the probability of drawing a black sock first and then a red sock. This involves calculating the probability of each event separately and then combining them. This approach ensures a clear understanding of how each draw contributes to the final probability.

1. Probability of Drawing a Black Sock First

The first step is to determine the probability of Luka drawing a black sock on his first attempt. To do this, we need to consider the number of black socks and the total number of socks in the bag. There is 1 black sock, and there are 5 socks in total (3 red + 1 white + 1 black). Using the basic probability formula:

P(Black) = (Number of Black Socks) / (Total Number of Socks)

P(Black) = 1 / 5

Therefore, the probability of drawing a black sock on the first draw is 1/5 or 0.2.

2. Probability of Drawing a Red Sock Second

Next, we need to calculate the probability of drawing a red sock on the second draw. Since Luka replaces the first sock, the total number of socks and the number of red socks remain the same as they were initially. This is a crucial detail, as it makes the two events independent. There are 3 red socks, and there are still 5 socks in total. The probability of drawing a red sock is:

P(Red) = (Number of Red Socks) / (Total Number of Socks)

P(Red) = 3 / 5

Thus, the probability of drawing a red sock on the second draw is 3/5 or 0.6.

3. Combining the Probabilities

Since the two events are independent (because the sock is replaced), we can find the probability of both events occurring in sequence by multiplying their individual probabilities. This is a fundamental rule in probability theory: for independent events A and B, the probability of both A and B occurring is:

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

In our case, event A is drawing a black sock, and event B is drawing a red sock. We have already calculated:

P(Black) = 1 / 5

P(Red) = 3 / 5

Now, we multiply these probabilities:

P(Black, then Red) = P(Black) * P(Red)

P(Black, then Red) = (1 / 5) * (3 / 5)

P(Black, then Red) = 3 / 25

Therefore, the probability of drawing 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%).

Alternative Approaches

While the step-by-step method provides a clear understanding of the probability calculation, there are alternative approaches to solving this problem. One such approach involves visualizing the possible outcomes using a probability tree. A probability tree is a diagram that represents all possible outcomes of a sequence of events along with their associated probabilities. It can be a helpful tool for understanding and solving more complex probability problems, especially those involving multiple stages or conditional probabilities. This visual representation can make the process more intuitive for some learners.

Using a Probability Tree

To construct a probability tree for this problem, we would start with the first draw. There are two branches emanating from the starting point: one for drawing a black sock (with a probability of 1/5) and another for not drawing a black sock (which includes drawing a red or white sock, with a combined probability of 4/5). From the “black sock” branch, we would then have two further branches for the second draw: one for drawing a red sock (probability 3/5) and another for not drawing a red sock (probability 2/5). Similarly, from the “not black sock” branch, we would have branches for drawing a red sock and not drawing a red sock. The probability of the sequence “black sock then red sock” is found by following the corresponding branches and multiplying the probabilities along the path:

(1/5) * (3/5) = 3/25

This method visually confirms our earlier calculation. While the probability tree may seem more complex for this relatively simple problem, it becomes invaluable when dealing with more intricate scenarios involving conditional probabilities or multiple stages.

Real-World Applications

The principles of probability, as demonstrated in this sock-drawing problem, have wide-ranging applications in various real-world scenarios. Understanding probability is crucial in fields such as finance, insurance, gambling, and even weather forecasting. In finance, for instance, investors use probability to assess the risk associated with different investment options. They might calculate the probability of a stock increasing or decreasing in value based on historical data and market trends. This helps them make informed decisions about where to allocate their capital. In the insurance industry, actuaries use probability to estimate the likelihood of various events, such as accidents, illnesses, or natural disasters. These estimates are then used to set insurance premiums, ensuring that the insurance company can cover potential payouts while remaining profitable.

In the world of gambling, probability is at the heart of every game. The odds of winning in a lottery, casino game, or sports bet are all based on probability calculations. Understanding these probabilities can help individuals make more informed decisions about their gambling behavior. For example, knowing the house edge in a casino game can help players choose games with better odds. Weather forecasting also relies heavily on probability. Meteorologists use weather models and historical data to estimate the probability of rain, snow, or other weather events. These probabilities are often expressed as percentages in weather forecasts, giving people an idea of the likelihood of certain weather conditions.

Conclusion

In conclusion, the problem of calculating the probability of drawing a black sock and then a red sock from a bag illustrates fundamental concepts in probability theory. By breaking down the problem into manageable steps, we can clearly see how to calculate the probability of independent events occurring in sequence. We first determined the probability of drawing a black sock (1/5), then the probability of drawing a red sock (3/5), and finally, we multiplied these probabilities to find the probability of both events occurring in order (3/25). This problem highlights the importance of understanding the principles of probability, which are applicable in various real-world scenarios, from finance and insurance to gambling and weather forecasting. By mastering these concepts, individuals can make more informed decisions and better understand the world around them.

This exercise not only reinforces the core concepts of probability but also demonstrates how these concepts can be applied to solve practical problems. The ability to calculate probabilities accurately is a valuable skill in many areas of life, and this sock-drawing problem serves as a simple yet effective illustration of this principle.