Question

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 are12 black socks, 8 red socks, and 16 brown socks in the drawer. None of the first three socks Said picks are the same color. What is the probability that he will not make a matched pair with his next pick? A. \(100 \%\) B. \(67 \%\) C. \(33 \%\) D. \(0 \%\)

Short answer

D. 0 %

Step-by-step solution

Understand the Problem

We need to find the probability that Said will not make a matched pair with his fourth sock pick, given that the first three socks picked are all different colors.

List Total Number of Socks

Total number of socks in the drawer is calculated as follows: \[ 12 \text{ black} + 8 \text{ red} + 16 \text{ brown} = 36 \text{ socks} \]

Socks Already Picked

Said has already picked 3 socks, each of a different color. For example, one black, one red, and one brown.

Calculate Remaining Socks

After picking one sock of each color, the remaining socks are: \[ 11 \text{ black} + 7 \text{ red} + 15 \text{ brown} = 33 \text{ socks} \]

Determine Non-Match Probability

To avoid a match, Said's fourth sock pick must be of a different color from the three he already has. However, he only has three possible colors, and he has already picked one of each. As soon as he picks another sock, it will match one of the colors he already has. Thus, the probability is given by \[ \frac{0}{33} = 0 \%\]

Key concepts

Statistics and Probability

Statistics is a major branch of mathematics that deals with collecting, analyzing, interpreting, and presenting data. One of the critical concepts in statistics is probability, which measures the likelihood of a certain event occurring.
Probability is calculated by dividing the number of successful outcomes by the total number of possible outcomes. For instance, in Said's sock problem, we want to find the probability that his fourth sock pick will not create a matched pair.
The total number of socks initially is 36 (black, red, and brown). Said picks three socks, each of a different color, reducing the total possible choices for the fourth sock to 33.
Since all three possible colors are already picked, any additional pick will be a match. Thus, the probability of not getting a matched pair is 0 out of 33.

Combinatorics in Probability

Combinatorics is the area of mathematics that focuses on counting, arranging, and finding patterns among sets of items. It plays a crucial role in solving probability problems.
In Said's problem, we apply combinatorics to determine the number of ways different outcomes can occur. Initially, Said has 12 black, 8 red, and 16 brown socks.

• Total socks = 12 + 8 + 16 = 36.
• He picks one sock each of black, red, and brown.
• Remaining socks = 11 black + 7 red + 15 brown = 33 total socks.

Understanding how to count these combinations helps us see that after picking one of each color, any further pick completes the matching process. Hence, there are no socks left to maintain a non-matching set, leading to a probability of 0 for avoiding a pair.

Mathematical Reasoning for Solutions

Mathematical reasoning involves using logical thought processes to solve mathematical problems. It includes understanding given conditions, making assumptions, and deriving conclusions logically.
Let's break down Said's sock problem using mathematical reasoning.

1. Said picks socks in the dark, one each of black, red, and brown.
2. We calculate the remaining socks: 11 black, 7 red, and 15 brown.
3. To avoid a match, his fourth pick should not be any of the already picked colors.
4. Given only three color options and one of each already picked, any new sock will match one of the original sets.
5. Thus, the probability of not making a matching pair is determined to be 0%.

Using reasoning to follow through each step ensures a clear understanding of why Said cannot avoid creating a matched pair.