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In probability theory, a sample space is the set of all possible outcomes or results that can occur in a given experiment. When considering the assignment of contracts to firms, each unique arrangement of assignments forms an individual outcome within the sample space.

To understand this better using our exercise, observe the process of assigning contracts to firms. For the first contract, any of the five firms (\(F_1\) to \(F_5\)) can be selected, each choice forming a possible outcome. With five firms available for the first contract (\(C_1\)), after assigning it, only four firms remain for the second contract (\(C_2\)), and finally three firms for the third contract (\(C_3\)).

The multiplicative principle of counting demonstrates the sample space size, calculated as:

• 5 choices for \(C_1\)
• 4 choices for \(C_2\)
• 3 choices for \(C_3\)

Therefore, the total number of possible combinations or sample points in this experiment is:\[5 \times 4 \times 3 = 60\]

Understanding the sample space helps visualize all potential ways contracts can be assigned, laying a foundation for calculating probabilities.

Combinatorics is a branch of mathematics concerning the counting, arrangement, and combination of elements within a set. It plays a crucial role in solving problems related to probabilities by determining how many possible arrangements or selections can be made.

In our contract-assignment scenario, combinatorics is applied to figure out how many different ways we can assign the contracts to the firms. By using the principle of permutations, which considers the order of assignments, we multiply the number of choices available at each stage:

• 5 options for the first contract (\(C_1\))
• After one firm is chosen, 4 options for the second (\(C_2\))
• Finally, 3 options for the third (\(C_3\))

Thus, the computation \(5 \times 4 \times 3 = 60\) tells us there are 60 unique ways to assign all the contracts.

Combinatorics simplifies the task of counting scenarios in complex operations, making it easier to determine probabilities when dealing with various assignments or selections.

In probability, conditional probability is the measure of the probability of an event occurring given that another event has already occurred. When attempting to find the probability of a firm receiving a contract, we consider each contract assignment conditionally.

Using firm \(F_3\) as an example, we are interested in evaluating the probability that this specific firm secures a contract. Let's break down the steps:

• If \(F_3\) is awarded \(C_1\), there are 12 ways to assign the other two contracts to the remaining firms.
• If \(F_3\) gets \(C_2\), similarly, there are 12 configurations for the others \((4 \times 3 = 12)\).
• If \(F_3\) receives \(C_3\), again 12 configurations emerge.

Adding these, we find 36 favorable outcomes for \(F_3\) attaining any contract.

Thus, the conditional probability that \(F_3\) secures a contract, given the total sample space of 60 possible assignments, is calculated as:\[\frac{36}{60} = \frac{3}{5}\]

This concept helps in understanding how the likelihood of an event changes when additional conditions or information are considered, a crucial concept in statistics and decision-making.