91Ó°ÊÓ

Let \(j\) and \(n\) be positive integers, with \(j \leq n\). An experiment consists of choosing, at random, a \(j\) -tuple of positive integers whose sum is at most \(n\). (a) Find the size of the sample space. Hint: Consider \(n\) indistinguishable balls placed in a row. Place \(j\) markers between consecutive pairs of balls, with no two markers between the same pair of balls. (We also allow one of the \(n\) markers to be placed at the end of the row of balls.) Show that there is a \(1-1\) correspondence between the set of possible positions for the markers and the set of \(j\) -tuples whose size we are trying to count. (b) Find the probability that the \(j\) -tuple selected contains at least one 1 .

Step by step solution

01

Find Total Number of Objects

Consider the problem as allocating **n** indistinguishable balls among **j+1** sections. Each section contains non-negative integers representing the values of the **j**-tuple and leftover section. We need to place **j** markers among the **n** balls to create these sections.

02

Apply Combinatorial Concepts

The total available positions for these markers are **n + j** because they can be placed between or at the ends of the **n** balls. The number of ways to choose **j** positions for these markers from **n+j** total positions is given by the combination formula: \( \binom{n+j}{j} \).

03

Interpret the Combination Result

The combination \( \binom{n+j}{j} \) gives us the size of the sample space, which is the total number of ways to choose the j-tuples such that their sum is at most **n**.

04

Find Tuples Containing at Least One 1

Now, note that if a tuple does not contain 1, then each entry in the **j**-tuple is at least 2. Therefore, redefine the tuple entries as **k**=tuple-1, transforming the task into finding numbers whose sum is at most **n-j** with **j** entries.

05

Calculate Restricted Sample Size

Apply the combination as before to find the number of **j**-tuples of non-negative integers that sum to at most **n-j** with the number of ways being \( \binom{n-1}{j-1} \).

06

Probability Calculation

Divide the number of favourable outcomes (having at least one 1) by the total number of outcomes found in Step 2: \( \frac{\binom{n-1}{j-1}}{\binom{n+j}{j}} \). Simplify this expression for the final probability.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Combinations

Combinations are a key concept in combinatorics that help us determine how many ways we can select a group of items from a larger set without regard to the order in which they are selected. This is different from permutations, which do take the order into account. In our problem, we use combinations to select positions for markers among a row of balls.

• Each position where a marker is placed represents a separation between tuple elements.
• The formula to calculate combinations is given by \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) where \( n \) is the total number of items, and \( r \) is the number of items to choose.
• In our case, the combination \( \binom{n+j}{j} \) gives us the number of ways to place \( j \) markers between the balls, creating different sections.

When we place these markers, we form groups of numbers adding up to at most \( n \). This is the essence of the problem, using combinations to partition integers.

Exploring the Sample Space

The sample space in probability is the set of all possible outcomes of an experiment. In our problem, the sample space consists of all possible \( j \)-tuples of positive integers that add up to at most \( n \).

• Visualize this by considering \( n \) indistinguishable balls and placing them in line with markers.
• The positions of these markers define the boundaries of our tuple elements and consequently the entire sample space.

The size of this sample space is crucial for calculating probabilities, as it counts how many outcomes fit within our specified rules. This size is determined by the number of ways to arrange the markers: \( \binom{n+j}{j} \).

Delving into Probability Calculation

To find the probability of an event, we compare the number of favorable outcomes to the total number of possible outcomes in the sample space. This is expressed as \[ P(A) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \]For our specific problem, we are interested in the probability that a chosen \( j \)-tuple contains at least one 1.Let's break this down:

• First, we identified the total number of \( j \)-tuples using the sample space size: \( \binom{n+j}{j} \).
• Then, to find the number of tuples having at least one 1, we counted those that don't (where each element is at least 2) and subtracted this number from the total.

This gave us the formula \( \frac{\binom{n-1}{j-1}}{\binom{n+j}{j}} \) to represent the probability required.

Decoding Tuples and Their Role

A tuple is an ordered list of elements. In mathematics, tuples can contain any number of elements, and they play a critical role in illustrating how elements can be grouped or separated.

• Tuples in our exercise are defined by their elements adding up to at most \( n \).
• When using the terms like \( j \)-tuples, it specifies that the tuple consists of \( j \) elements.

Understanding tuples helps us recognize that although the elements can be viewed as distinct, in our problem, they emerge from partitioning indistinguishable objects (balls) into distinguishable parts (tuple values). This understanding is essential when solving combinatoric problems like ours, where the arrangement impacts the outcome.

Unpacking Combinatorial Probability

Combinatorial probability uses combinatorial techniques to calculate the likelihood of outcomes. This approach is particularly useful when dealing with large, complex sample spaces like in our problem.

• With combinatorial probability, we count possible outcomes directly, using combinations and other counting principles.
• For instance, we used combinations to define the sample space and calculate probabilities of particular outcomes (tuples with at least one 1).

By structuring the problem into understandable parts - counting marker positions, constructing tuples, identifying relevant outcomes - we apply combinatorial probability effectively.