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In probability theory, understanding independent events is crucial for analyzing how separate occurrences can affect overall outcomes.
When two events are independent, it means that the outcome of one event does not affect the outcome of the other. In mathematical terms, two events, event A and event B, are said to be independent if the probability of both events happening simultaneously is equal to the product of their individual probabilities. This can be expressed as:

• \( P(A \text{ and } B) = P(A) \cdot P(B) \)

In the context of the exercise with hunters A and B, their shots at the target are considered independent. That means whether or not hunter A hits the target does not affect the probability of hunter B hitting the target.
Therefore, to analyze the problem's solution, each resulting situation (zero, one, or two hits) can be calculated individually using their independent probabilities.

Probability equations help us express the likelihood of different outcomes in a quantifiable manner. In this particular exercise, we start by defining the events:

• The chance of zero hits, when neither A nor B hits the target, is derived from the probabilities of missing the target for both: \((1 - p_{1}) \cdot (1 - p_{2})\).
• The chance of one hit involves two scenarios, either hunter A hits while B misses, or B hits while A misses, which gives us: \((1 - p_{1}) \cdot p_{2} + p_{1} \cdot (1 - p_{2})\).
• Lastly, the probability of both hitting the target (two hits) is simply: \(p_{1} \cdot p_{2}\).

These equations form the base for further calculations by setting them equal to determine possible values of \(p_{1}\) and \(p_{2}\) that satisfy equality among all outcomes.
Such equations allow us to answer more complex questions by equating different scenarios and solvable expressions.

When dealing with multiple probability equations at once, combining them into a system of equations can streamline the solution process. A system of equations allows mathematicians to solve multiple equations simultaneously for common variables.
In the exercise provided, we must solve a system involving three equations based on different probability outcomes:

• \((1 - p_{1}) \cdot (1 - p_{2}) = x\)
• \((1 - p_{1}) \cdot p_{2} + p_{1} \cdot (1 - p_{2}) = x\)
• \(p_{1} \cdot p_{2} = x\)

By setting these equations equal to a common variable \(x\), we examine the conditions under which each equation balances equally with the others.
Throughout the exercise, solving these equations shows if particular values for \(p_{1}\) and \(p_{2}\) allow all three scenarios to have identical probabilities.
Ultimately, the conclusion is reached by checking the conditions under which these probabilities equalize, thereby validating if such \(p_{1}\) and \(p_{2}\) values exist.