Problem 15 - 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 has been drawn at random from the drawer, what is the probability that is red? (b) Once a sock has been 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

The probability of the first red sock is \(\frac{1}{3}\); for a second red sock, it's \(\frac{5}{17}\).

Step 1: Analyze the Total Number of Socks

First, calculate the total number of socks in the drawer. There are 3 pairs of red socks, 2 pairs of white socks, and 4 pairs of black socks. Each pair contains 2 socks.\[\text{Total socks} = (3 + 2 + 4) \times 2 = 18\]

Step 2: Calculate the Probability of Drawing One Red Sock

We need to find the probability of drawing one red sock from the total of 18 socks. Since there are 3 pairs of red socks, there are \(3 \times 2 = 6\) red socks in total.\[P(\text{Red sock}) = \frac{6}{18} = \frac{1}{3}\]

Step 3: Adjust for Drawing the Second Red Sock

Given that the first sock drawn is red, the focus narrows on the remaining socks. We have drawn one red sock, so there are now 5 remaining red socks and 17 socks in total.\[P(\text{Second Red sock | First Red sock}) = \frac{5}{17}\]

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, combination, and permutation of objects. It is essential for calculating probabilities in many scenarios. In this exercise, combinatorics helps us determine how many socks of each color are available in the drawer.
Understanding combinations is crucial when dealing with situations where order does not matter, while permutations come into play when the order is significant.

• For instance, knowing there are 3 pairs of red socks allows us to calculate the total red socks, which is 6.
• Similarly, 2 pairs of white and 4 pairs of black socks translate to 4 and 8 socks, respectively.

This foundational analysis of the number of available socks is necessary for calculating the probabilities of drawing a particular colored sock, as illustrated in the first task of this exercise, where the probability of drawing a red sock is calculated from the total number of socks.

Conditional Probability

Conditional probability is used when the outcome of an event is affected by the occurrence of a previous event. It helps in calculating the probability of the next event, given that a specific event has occurred. In the case of this exercise, once a red sock is drawn, the probability of drawing another red sock to form a pair becomes a conditional probability problem.
The key formula for conditional probability is:
\[P(A | B) = \frac{P(A \cap B)}{P(B)}\]
In our example, after drawing a red sock first, the calculation changes because the total number of socks, and the number of red socks, are different from the initial conditions.

• Using conditional probability, the number of remaining red socks becomes 5, with the total socks being 17.
• This adjusts the probability for the second event to \(\frac{5}{17}\).

Understanding conditional probability allows us to correctly calculate these chances, providing insights into how probabilities shift with new information.

Probability Theory

Probability theory is a fundamental principle that quantifies uncertainty. It gives a mathematical framework to predict the likelihood of various outcomes. This framework is used extensively in calculating the probability of drawing a specific sock from the drawer.
In probability theory, probabilities range from 0 to 1. A probability of 0 means the event cannot occur, while a probability of 1 means the event is certain to occur.
For our exercise:

• The probability of drawing one red sock initially is calculated as \(\frac{1}{3}\) since there are 6 red socks out of a total of 18 socks.
• This basic idea of totaling up the favorable outcomes over the total possible outcomes underpins much of probability computations.

Probability theory includes other concepts like independence, random variables, and expected value, although they are not directly addressed in this simple drawer exercise. Nevertheless, understanding the basic principles allows us to expand into more complicated problems efficiently.