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When dealing with probability, one important concept is that of independent events. Independent events are those where the occurrence of one event does not affect the occurrence of another. For example, if Bill and Mike are each trying to solve a problem on their own, the success or failure of Bill has no impact on Mike's chances of success.
This independence is crucial because it allows us to easily calculate probabilities for combined scenarios. In our problem, if the probability that Bill solves a problem is \(p_1\) and the probability that Mike solves it is \(p_2\), independence means their joint probability is simply the product of their individual probabilities: \(p_1 \times p_2\). This formula will be very important as we explore more complex probability calculations involving unions and intersections of events.

The probability of the union of two events is a fundamental concept in probability theory. It refers to the probability that at least one of the events occurs. For two events A and B, the union is represented as \(A \cup B\). This can be thought of as the combined probability that either event A occurs, event B occurs, or both occur.
In our example with Bill and Mike, the formula used is:

• \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

Here, \(P(A)\) is the probability of Bill solving the problem and \(P(B)\) is that of Mike solving it. The subtraction of \(P(A \cap B)\) ensures we do not double count the scenario where both solve the problem, which is represented by the intersection.

Using a mathematical proof helps us validate probability statements. To show the probability expression given in the exercise, we start by using known formulas and logical steps.
First, recognize that Bill and Mike working independently means we can use the multiplication rule for the intersection of two independent events: \(P(A \cap B) = p_1 \times p_2 \).
Then, applying the probability of union formula, \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\), we substitute the values:

• \(P(A \cup B) = p_1 + p_2 - p_1 \times p_2\)

This step-by-step substitution confirms that the derived formula \(p_1 + p_2 - p_1 p_2\) indeed represents the probability that at least one of them can solve the problem.

The concept of event intersection is another key idea in probability theory. The intersection of two events represents the scenario where both events occur simultaneously. It is denoted as \(A \cap B\).
In our example, the intersection refers to both Bill and Mike successfully solving the problem. Since their actions are independent, we calculate this as \(p_1 \times p_2\). This makes intuitive sense: the likelihood of both succeeding is the product of their individual probabilities.
Understanding intersections is vital when determining the overall probability of combined events. Not only does it help in calculating the union of probabilities, but it also helps avoid errors like double counting scenarios where both events happen, thus making the calculations accurate.