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Probability is the study of how likely an event is to happen. In our case, it refers to the assignment of jobs to processors in a computing center. When we say all assignments are equally likely, we imply that each possible assignment has the same chance of occurring. This means that for

• any specific assignment of jobs to processors,
• or any particular configuration of job distribution,

all are uniform in likelihood. The number of possible ways to assign jobs is determined by the number of processors raised to the power of the number of jobs. For example, if there are 3 processors and each job can be assigned to any of them, mathematically, \[ 3^n \] where \( n \) is the number of jobs. Understanding probability helps us determine the chance of an event. Here, the event is that exactly one processor ends up with no jobs. By calculating the ratio of the successful configurations (favorable outcomes) to the total possible configurations, we obtain the probability of this event occurring.

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. In our problem, it helps us figure out how many ways we can assign jobs to processors in different configurations. When we talk about assigning jobs randomly to processors,

• combinatorics helps determine the number of valid assignments,
• and how they can be distributed to satisfy specific conditions.

For instance, to satisfy the condition that one processor ends up with no jobs, we need to count the number of ways to assign \( n \) jobs to the remaining two processors. This boils down to basic counting rules:1. Choosing which processor out of the three will have no jobs: 3 ways.2. Assigning the \( n \) jobs to the other two processors: \( 2^n \) ways (because each job can be assigned to either one of the two remaining processors). Hence, through combinatorics, we determine that there are \( 3 imes 2^n \) favorable outcomes meeting our criteria, providing insights into how specific distributions of jobs can occur.

Random assignment involves distributing elements (in this case, jobs) across various entities (processors), with the key feature that each distribution is independent of the others and all are equally likely. This process is akin to drawing names from a hat, ensuring each name has an equal chance of being picked each time.In the problem scenario:

• Each job is randomly and independently assigned to one of the 3 processors.
• Every possible distribution of these jobs is initially considered equally probable.

Utilizing random assignment ensures there is no bias or predetermined pattern in how jobs are allocated. However, since our goal is for one processor to have no jobs, the nature of random assignment is crucial. Calculating the probability of a specific configuration involves understanding this randomness, as we consider each of the possible configurations (\( 3^n \) configurations) and isolate those where one processor ends up without jobs. Confirming distributions follow our desired pattern means relying on this randomness algorithm to favorably align with specific configurations, using a probability-based approach.