Probability: Drawing Socks Without Replacement

by Dimemap Team 47 views

Let's dive into a classic probability problem: figuring out the chances of pulling out two white socks in a row from a drawer, without putting the first one back. It sounds simple, but it involves a key concept called conditional probability. So, grab your thinking caps, and let's get started!

Understanding the Sock Drawer Scenario

Okay, imagine you've got a drawer overflowing with loose socks. Inside, there are 2 red socks, 2 green socks, and 6 white socks. That's a total of 10 socks. Now, you're going to reach in and grab one sock, hoping it's white. Then, without replacing it, you're going to grab another one, again hoping for white. The question is, how do we calculate the probability of this happening?

To really get a handle on this, we need to break it down into steps and understand how each draw affects the odds for the next. This is where the idea of "without replacement" becomes super important. When you don't put the sock back, you're changing the total number of socks and the number of white socks remaining, which changes the probabilities. Let's explore this concept with an example, what would be the probability if we added 10 blue socks, how would that change the final probability result?

Step 1: Probability of the First White Sock

Initially, we have 6 white socks out of a total of 10 socks. So, the probability of drawing a white sock on your first try is simply the number of white socks divided by the total number of socks.

Probability (First Sock is White) = (Number of White Socks) / (Total Number of Socks) = 6 / 10 = 3 / 5 = 0.6 or 60%

So, you have a 60% chance of grabbing a white sock right off the bat. Not bad, right?

Step 2: Probability of the Second White Sock (Given the First Was White)

Here's where it gets interesting. Let's say you successfully pulled out a white sock on your first try and didn't put it back. Now, how many socks are left in the drawer? Well, there were 10, and you took one out, so there are now 9 socks remaining. And how many of those are white? Since you already took out one white sock, there are only 5 white socks left.

This means the probability of drawing a second white sock depends on the fact that you already drew a white sock. This is called conditional probability. We calculate it like this:

Probability (Second Sock is White | First Sock was White) = (Number of Remaining White Socks) / (Total Number of Remaining Socks) = 5 / 9 ≈ 0.556 or 55.6%

So, after successfully grabbing a white sock the first time, your chance of grabbing another white sock drops to about 55.6%.

Step 3: Combining the Probabilities

Now, to find the probability of both events happening (drawing a white sock, then another white sock without replacement), we need to multiply the probabilities of each individual event.

Probability (White, then White) = Probability (First Sock is White) * Probability (Second Sock is White | First Sock was White)

Probability (White, then White) = (6 / 10) * (5 / 9) = 30 / 90 = 1 / 3 ≈ 0.333 or 33.3%

Therefore, the probability of pulling out a white sock, not replacing it, and then pulling out another white sock is approximately 33.3%.

Why This Matters: Understanding Conditional Probability

This sock example perfectly illustrates the concept of conditional probability. Conditional probability is the probability of an event occurring given that another event has already occurred. The notation P(B|A) represents the probability of event B happening given that event A has already happened. In our sock example, event A is drawing a white sock on the first draw, and event B is drawing a white sock on the second draw.

Conditional probability is used everywhere in real life, from medical diagnoses to financial risk assessment. For example, a doctor might assess the probability of a patient having a disease given that they have certain symptoms. Or, a bank might assess the probability of a borrower defaulting on a loan given their credit history.

Key Takeaways

  • Without Replacement: When you don't replace an item, the probabilities for subsequent events change.
  • Conditional Probability: The probability of an event can depend on whether another event has already occurred.
  • Multiplying Probabilities: To find the probability of multiple events happening in sequence, multiply their individual probabilities (taking into account any conditional probabilities).

Let's Consider More Sock Scenarios

To really solidify your understanding, let's explore some variations of our sock problem.

Scenario 1: Drawing Two Socks of the Same Color (Any Color)

What if we wanted to know the probability of drawing two socks of the same color, regardless of what color that is? This is a bit more complex, but we can break it down. We need to consider the probability of drawing two red socks, two green socks, or two white socks, and then add those probabilities together.

  • Probability (Two Red Socks): (2/10) * (1/9) = 2/90
  • Probability (Two Green Socks): (2/10) * (1/9) = 2/90
  • Probability (Two White Socks): (6/10) * (5/9) = 30/90

Probability (Two Socks of the Same Color) = (2/90) + (2/90) + (30/90) = 34/90 = 17/45 ≈ 0.378 or 37.8%

Scenario 2: Drawing a Red Sock, then a Green Sock

What if we wanted to know the probability of drawing a red sock first, and then a green sock (without replacement)?

  • Probability (Red Sock First): 2/10
  • Probability (Green Sock Second | Red Sock First): 2/9 (since there are still 2 green socks left out of 9 total)

Probability (Red, then Green) = (2/10) * (2/9) = 4/90 = 2/45 ≈ 0.044 or 4.4%

Common Mistakes to Avoid

  • **Forgetting to Adjust for