Question

Use the following for questions 6–8. Said is getting dressed without turning on the light so that he won’t wake his brother. He is picking socks, but he cannot tell the colors of the socks in his sock drawer in the dark. He knows that there are 12 black socks, 8 red socks, and 16 brown socks in the drawer. If Said picks out a brown sock first, what is the probability that he will need to make at least three picks to get a pair? A. \(\frac{3}{4}\) B. \(\frac{4}{7}\) C. \(\frac{4}{9}\) D. \(\frac{1}{3}\)

Short answer

The probability that Said will need at least three picks to get a pair is \( \frac{55}{63} \).

Step-by-step solution

- Understanding the Total Number of Socks

First, determine the total number of socks in the drawer. There are 12 black socks, 8 red socks, and 16 brown socks. In total, there are: \[ 12 + 8 + 16 = 36 \text{ socks} \]

- Probability of First Sock Being Brown

Calculate the probability that Said picks a brown sock first. There are 16 brown socks out of a total of 36 socks. Thus, the probability is: \[ \frac{16}{36} = \frac{4}{9} \]

- Second Sock After Picking Brown First

After picking a brown sock first, there are now 35 socks left (12 black, 8 red, 15 brown). If Said is to get a pair, he needs to draw another brown sock. There are 15 brown socks remaining out of the 35 total socks, thus the probability for this is: \[ \frac{15}{35} = \frac{3}{7} \]

- Probability of Not Getting a Brown Sock Second

Calculate the probability that the second sock picked is not brown. Since there are 35 socks left after picking one brown, and out of these, 20 are not brown (12 black, 8 red): \[ 1 - \frac{3}{7} = \frac{4}{7} \]

- Probability of Getting a Pair of Different Color Socks in the Third Pick

Now, looking at the third pick, Said must pick a brown sock to make a pair. The total ways he can pick a brown sock first, then a non-brown sock second, and lastly another brown sock third is determined by multiplying the respective probabilities: \[ \frac{4}{9} \times \frac{4}{7} \times \frac{14}{34} = \frac{4}{9} \times \frac{4}{7} \times \frac{7}{17} = \frac{4}{9} \times \frac{4}{7} \times \frac{1}{2} = \frac{16}{126} = \frac{8}{63} \]

- Simplified Final Probability

To summarize, the probability that Said will need to draw at least three socks to get a pair is: \[ 1 - \frac{8}{63} = \frac{55}{63} \]

Key concepts

Probability

Probability is the measure of how likely an event is to occur. It ranges from 0 (impossible) to 1 (certain). In this problem, Said is trying to pick socks in the dark, not knowing their colors. To solve such problems, we need to calculate the likelihoods of various events happening.
Let's recall here that probability is calculated as:
\( P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}} \)
In Said's case, he needs to draw at least three socks to get a pair. We calculate these probabilities step by step, considering each scenario and available options.

Combinatorics

Combinatorics is a field in mathematics concerned with counting, arrangement, and combination of objects.
In our problem, Said's task of drawing socks without seeing their color involves concepts from combinatorics. We first calculate the total number of socks, then determine probabilities for each drawing event. For example:
First, the total number of socks is calculated as:
\( 12 + 8 + 16 = 36 \)
Next, we calculate the probability of picking a brown sock first, then a non-brown sock second, and so forth.
This step-by-step calculation involves arranging the socks into potential groups to find the likelihood of Said needing at least three picks.

GED Math

The General Educational Development (GED) test includes probability and combinatorics questions similar to Said’s sock-drawing problem.
Understanding this problem helps in preparing for such questions efficiently. We start by identifying the total number of socks available and then see how the probabilities change with each pick.
These skills involve basic arithmetic and understanding fractions, essential for the GED math section. Practice with step-by-step solutions can help build confidence and improve problem-solving speed.
By breaking down each step, as shown in the solution, students can see the logical progression and the importance of each calculation in determining the final probability.

Educational Mathematics

Educational Mathematics is all about understanding the concepts to solve problems like this one.
This specific exercise teaches students how to apply probability and combinatorics practically. Each step in the solution builds on previous knowledge - from calculating the total number of items (socks), to finding the probabilities of specific outcomes.
For example:

• First, finding the probability of picking a brown sock.
• Then, recalculating probabilities as the total number of socks changes with each pick.

This practical application helps cement understanding and equips students with tools to tackle similar real-world problems.
It's essential to understand the foundational principles, as they provide a systematic way to approach and solve complex problems, enhancing overall mathematical literacy.