Socks in a Drawer A drawer contains 11 identical red socks...

Socks in a Drawer A drawer contains 11 identical red socks and 8 identical black socks. Suppose that you choose 2 socks at random in the dark. $$ \begin{array}{l}{\text { a. What is the probability that you get a pair of red }} \\ {\text { socks? }} \\ {\text { b. What is the probability that you get a pair of black }} \\ {\text { socks? }} \\ {\text { c. What is the probability that you get 2 unmatched }} \\ {\text { socks? }} \\ {\text { d. Where did the other red sock go? }}\end{array} $$

Socks in a Drawer A drawer contains 11 identical red socks and 8 identical black socks. Suppose that you choose 2 socks at random in the dark.
$$
\begin{array}{l}{\text { a. What is the probability that you get a pair of red }} \\ {\text { socks? }} \\ {\text { b. What is the probability that you get a pair of black }} \\ {\text { socks? }} \\ {\text { c. What is the probability that you get 2 unmatched }} \\ {\text { socks? }} \\ {\text { d. Where did the other red sock go? }}\end{array}
$$

Key Concepts

Combinatorial Analysis

This concept involves counting the number of ways events can occur. In problems like these, combinatorial analysis is used to determine how many pairs of socks can be drawn from a set without caring about the order. It forms the basis for calculating probabilities by identifying both the total number of outcomes and the subset of outcomes that meet a specific condition.

Probability Theory

Probability theory enables us to quantify the likelihood of events occurring under uncertainty. In the context of drawing socks from a drawer, it is used to compute the chance of selecting two socks of a particular color or one sock of each color by comparing the number of favorable outcomes to the total number of possible outcomes.

Sampling Without Replacement

Sampling without replacement is a scenario where each item drawn is not returned to the set before the next draw, which means the probabilities change with each draw. This is crucial in problems involving draws from a finite set, such as socks in a drawer, where the outcome of the first draw affects the outcome of the second draw.

Favorable Outcomes Versus Total Outcomes

A fundamental principle in probability is comparing the number of favorable outcomes (those that satisfy the event in question) to the total number of outcomes in the sample space. Calculating these values using combinatorial methods allows one to determine the event’s probability, as illustrated by comparing the number of specific sock pairs to all possible pairs.

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