Problem 47 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Suppose the probability that Bill can solve a problem is \(p_{1}\) and the probability that Mike can solve it is \(p_{2}\). Show that the probability that Bill and Mike working independently can solve the problem is \(p_{1}+p_{2}-p_{1} p_{2}\).

Step-by-Step Solution

Answer

The probability that Bill and Mike working independently can solve the problem is \(p_1 + p_2 - p_1 p_2\), which represents the probability that at least one of them solves the problem. This result is obtained by applying the formula for the probability of the union of two events: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\), where event A represents Bill solving the problem, event B represents Mike solving the problem, and \(P(A \cap B) = p_1 * p_2\).

Step 1: Identify the events and their probabilities

Event A (Bill solving the problem) has a probability of \(p_1\), while event B (Mike solving the problem) has a probability of \(p_2\). As they are working independently, the probability of both events happening at the same time (A and B, or A ∩ B) is equal to \(p_1 * p_2\).

Step 2: Apply the union formula

We are now going to use the formula for the probability of the union of two events: \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\] Since \(P(A \cap B) = p_1 * p_2\), we can substitute it into the formula: \[P(A \cup B) = p_1 + p_2 - p_1 p_2\]

Step 3: Interpret the result

The expression \(P(A \cup B)\) represents the probability that at least one of them solves the problem. So, the probability that Bill and Mike working independently can solve the problem is \(p_1 + p_2 - p_1 p_2\).

Key Concepts

Independent Events

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.

Probability of Union

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.

Mathematical Proof

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.

Event Intersection

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.