Problem 9 - Algebra and Trigonometry

Question

A drawer contains an unorganized collection of 18 socks— three pairs are red, two pairs are white, and four pairs are black. (a) If one sock is drawn at random from the drawer, what is the probability that it is red? (b) Once a sock is drawn and discovered to be red, what is the probability of drawing another red sock to make a matching pair?

Step-by-Step Solution

Answer

(a) \(\frac{1}{3}\); (b) \(\frac{5}{17}\).

Step 1: Determine Total Number of Socks

First, calculate the total number of socks. Each pair of socks consists of 2 socks. Thus, for 3 pairs of red socks, we have 3 \(\times\) 2 = 6 red socks, for 2 pairs of white socks, we have 2 \(\times\) 2 = 4 white socks, and for 4 pairs of black socks, we have 4 \(\times\) 2 = 8 black socks. Adding these, the total number of socks is 6 + 4 + 8 = 18 socks.

Step 2: Calculate Probability of Drawing a Red Sock

The probability of drawing one red sock from the drawer can be calculated as the number of red socks divided by the total number of socks. Hence, the probability is \(\frac{6}{18}\), which simplifies to \(\frac{1}{3}\).

Step 3: Adjust Sock Count after Drawing a Red Sock

Once a red sock is drawn, the total number of socks becomes 17. Additionally, there are now 5 red socks remaining instead of 6, since one red sock has already been drawn.

Step 4: Calculate Probability of Drawing Another Red Sock

Now, calculate the probability of drawing another red sock. This is given by the number of remaining red socks divided by the remaining total number of socks. Therefore, it is \(\frac{5}{17}\).

Key Concepts

Probability of Drawing Red Socks

To understand how likely it is to pick a red sock from an unorganized sock collection, we turn to the concept of probability. Probability is like a measure that helps us understand how often we might expect an event to happen if we repeated an action many times. In this scenario, the event is drawing a red sock from the drawer.Let's think about our drawer. Inside it, we have 18 socks total. Of those, 6 socks are red. To find the chance, or probability, of grabbing a red sock, we use the formula:\[ \text{Probability of red sock} = \frac{\text{Number of red socks}}{\text{Total number of socks}} \]Plugging in our numbers, that's \( \frac{6}{18} \), which simplifies down to \( \frac{1}{3} \). So, each time you reach into that drawer, there's a one-third chance you'll end up holding a red sock.

Unorganized Socks Collection

An unorganized collection can sometimes feel chaotic, but it also provides us with an intriguing exploration of probability. Here, our socks are all mixed up in a drawer, with no sense of order. We have:

• 6 red socks (from 3 pairs)
• 4 white socks (from 2 pairs)
• 8 black socks (from 4 pairs)

This means that at any given time, the likelihood of pulling one particular color is not influenced by any order. In an unorganized setting, each sock is just as available to be picked as any other, making probability calculations straightforward. When dealing with probabilities in unorganized collections, it is important to fully account for the total and specific groups involved, just like the number of red, white, and black socks here. This ensures each probabilistic determination accurately reflects possible outcomes relative to the total.

Step by Step Probability Calculation

Step-by-step calculations take the mystery out of probability exercises by breaking down each part into manageable pieces. Here's how we go about it for our sock-drawer example:**Step 1: Calculate Total**The first thing we do is to count up the total number of socks. Here, we had 18 socks—a mix of red, white, and black specified by their pairs.**Step 2: Compute Initial Probability**Then, we looked for the probability of the first event, drawing a red sock. This was found by comparing the number of red socks, 6, to the total, 18, resulting in \( \frac{6}{18} = \frac{1}{3} \).**Step 3: Adjust Total After Draw**Once a red sock was drawn, it changed the remaining count. Now, we have only 17 left, and only 5 red socks because one was drawn already.**Step 4: Compute New Probability**Finally, we calculated the new probability of drawing another red sock from the adjusted total. This turned out to be \( \frac{5}{17} \), since there are 5 red socks left out of a new total of 17. By walking through these steps, you understand not just the answers, but the process of probability as it applies in simple cases like these.