Problem 9
Question
A set \(S\) contains 7 elements. A non-empty subset \(A\) of \(S\) and an element \(\mathrm{x}\) of \(\mathrm{S}\) are chosen at random. Then the probability that \(x \in A\) is: \(\quad\) Online April 11, 2014] (a) \(\frac{1}{2}\) (b) \(\frac{64}{127}\) (c) \(\frac{63}{128}\) (d) \(\frac{31}{128}\)
Step-by-Step Solution
Verified Answer
The probability is \(\frac{63}{127}\).
1Step 1: Determine the Total Number of Non-Empty Subsets
For a set with 7 elements, the total number of subsets is given by \(2^7\). This includes the empty set as well. Thus, there are \(2^7 - 1 = 127\) non-empty subsets of \(S\).
2Step 2: Count Subsets Containing Element x
If \(x\) is an element of \(S\), any subset that includes \(x\) can be formed by picking additional elements from the remaining 6 elements. Here, the total number of such subsets is given by \(2^6\).
3Step 3: Calculate Probability of x in Subset A
With \(63\) subsets including the element \(x\) and \(127\) total non-empty subsets, the probability that a randomly chosen subset contains \(x\) is \(\frac{63}{127}\).
4Step 4: Present the Final Answer
After comparing the options, the probability that \(x \in A\) is \(\frac{63}{127}\).
Key Concepts
Set TheoryCombinatoricsSubsets
Set Theory
Set theory is a fundamental concept in mathematics that deals with the study of sets, which are collections of objects. These objects can be anything from numbers to letters or even more complex entities.
A set is usually denoted by capital letters such as \( S \), and the objects within it are called elements. For example, if \( S = \{ 1, 2, 3, 4, 5, 6, 7 \} \), then \( 1 \), \( 2 \), and so forth up to \( 7 \), are elements of the set.
In set theory, the concept of subsets is also important. A subset is simply a set that contains some or all elements of another set. Every set is a subset of itself, and the empty set, which contains no elements, is a subset of every set. This exercise explores the total number of subsets of a set—a key facet of set theory.
Knowing about set theory is critical when learning about probability and other advanced topics in mathematics. It allows you to better understand how different collections of objects can interact and be organized.
A set is usually denoted by capital letters such as \( S \), and the objects within it are called elements. For example, if \( S = \{ 1, 2, 3, 4, 5, 6, 7 \} \), then \( 1 \), \( 2 \), and so forth up to \( 7 \), are elements of the set.
In set theory, the concept of subsets is also important. A subset is simply a set that contains some or all elements of another set. Every set is a subset of itself, and the empty set, which contains no elements, is a subset of every set. This exercise explores the total number of subsets of a set—a key facet of set theory.
Knowing about set theory is critical when learning about probability and other advanced topics in mathematics. It allows you to better understand how different collections of objects can interact and be organized.
Combinatorics
Combinatorics is a branch of mathematics that focuses on counting, arrangement, and combination of objects within a set. It is essential for solving problems related to the organization and selection of distinct items or elements.
In the context of this problem, combinatorics helps in determining the total number of subsets that can be formed from a set. For any given set with \( n \) elements, the total number of subsets (including the empty set) is \( 2^n \). This is because each element can either be included or excluded from a subset, leading to two possibilities per element.
Therefore, with a set \( S \) having 7 elements, as given in the exercise, the total number of subsets is \( 2^7 \), which equals 128. Subtract the empty set, and you are left with 127 non-empty subsets.
In the context of this problem, combinatorics helps in determining the total number of subsets that can be formed from a set. For any given set with \( n \) elements, the total number of subsets (including the empty set) is \( 2^n \). This is because each element can either be included or excluded from a subset, leading to two possibilities per element.
Therefore, with a set \( S \) having 7 elements, as given in the exercise, the total number of subsets is \( 2^7 \), which equals 128. Subtract the empty set, and you are left with 127 non-empty subsets.
- Combinatorics relies heavily on principles like permutations and combinations.
- It plays a critical role in a wide range of fields, from mathematics to computer science.
Subsets
Subsets are a crucial concept in both set theory and combinatorics. A subset is a set made up of elements from another set, and it can be empty or non-empty. If set \( A \) is a subset of set \( B \), then every element of \( A \) is also an element of \( B \).
In the problem at hand, we are concerned with non-empty subsets of a set \( S \) that has 7 elements. Non-empty subsets are those that contain at least one element. The question asks us to compute the probability that a specific element \( x \) of \( S \) is included in a randomly chosen non-empty subset.
To find this, we count all non-empty subsets of \( S \) and then count those subsets which include the element \( x \). Using combinatorial techniques, we find that there are \( 2^6 = 64 \) subsets that contain \( x \), as the remaining elements offer \( 2^6 \) different combinations.
The probability of \( x \) being in a randomly selected non-empty subset is then formed by the ratio of these two quantities, \( \frac{63}{127} \), reflecting the number of subsets containing \( x \) against the total number of non-empty subsets. This demonstrates the power and utility of subset understanding in solving probabilistic problems.
In the problem at hand, we are concerned with non-empty subsets of a set \( S \) that has 7 elements. Non-empty subsets are those that contain at least one element. The question asks us to compute the probability that a specific element \( x \) of \( S \) is included in a randomly chosen non-empty subset.
To find this, we count all non-empty subsets of \( S \) and then count those subsets which include the element \( x \). Using combinatorial techniques, we find that there are \( 2^6 = 64 \) subsets that contain \( x \), as the remaining elements offer \( 2^6 \) different combinations.
The probability of \( x \) being in a randomly selected non-empty subset is then formed by the ratio of these two quantities, \( \frac{63}{127} \), reflecting the number of subsets containing \( x \) against the total number of non-empty subsets. This demonstrates the power and utility of subset understanding in solving probabilistic problems.
Other exercises in this chapter
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