Problem 82 - Applied Statistics and Probability for Engineers

Question

\(+\) If \(P(A)=0.3, P(B)=0.2,\) and \(P(A \cap B)=0.1,\) deter- mine the following probabilities: (a) \(P\left(A^{\prime}\right)\) (b) \(P(A \cup B)\) (c) \(P\left(A^{\prime} \cap B\right)\) (d) \(P\left(A \cap B^{\prime}\right)\) (e) \(P\left[(A \cup B)^{\prime}\right]\) (f) \(P\left(A^{\prime} \cup B\right)\)

Step-by-Step Solution

Answer

(a) 0.7, (b) 0.4, (c) 0.1, (d) 0.2, (e) 0.6, (f) 0.8

Step 1: Calculate Complement Probability P(A')

The probability of the complement of event A, denoted as \(P(A')\), is calculated as \(P(A') = 1 - P(A)\). Given \(P(A) = 0.3\), we have:\[ P(A') = 1 - 0.3 = 0.7 \]

Step 2: Use Inclusion-Exclusion for P(A ∪ B)

The probability of the union of events A and B is given by the formula \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Using the given probabilities, we calculate:\[ P(A \cup B) = 0.3 + 0.2 - 0.1 = 0.4 \]

Step 3: Calculate Probability P(A' ∩ B)

The probability \(P(A' \cap B)\) can be found using the relationship \(P(A' \cap B) = P(B) - P(A \cap B)\). Given the values:\[ P(A' \cap B) = 0.2 - 0.1 = 0.1 \]

Step 4: Calculate Probability P(A ∩ B')

The probability \(P(A \cap B')\) is calculated using \(P(A \cap B') = P(A) - P(A \cap B)\). Thus:\[ P(A \cap B') = 0.3 - 0.1 = 0.2 \]

Step 5: Find Probability of Complement P[(A ∪ B)']

The complement probability of \(P(A \cup B)\) is given by \(P[(A \cup B)'] = 1 - P(A \cup B)\). We found \(P(A \cup B) = 0.4\), so:\[ P((A \cup B)') = 1 - 0.4 = 0.6 \]

Step 6: Use De Morgan's Law for P(A' ∪ B)

Using the identity \(P(A' \cup B) = 1 - P(A \cap B')\), given \(P(A \cap B') = 0.2\), the calculation is:\[ P(A' \cup B) = 1 - 0.2 = 0.8 \]

Key Concepts

Complement Rule

The Complement Rule helps us find the probability of an event not occurring, also known as the complement of an event. If an event has a probability of occurring being the value of \(P(A)\), then the probability of the event not happening, \(P(A')\), is simply \(1 - P(A)\). This is because the total probability for all possible outcomes of a specific situation must sum to 1.

Consider event A with a probability of 0.3. Using the Complement Rule, \(P(A') = 1 - 0.3 = 0.7\). Hence, the chance that event A does not occur is 0.7. This rule is fundamental in probability theory, ensuring that we account for all potential scenarios by calculating what is sometimes termed 'the remainder' probability of an event.

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a powerful tool used to find the probability of the union of multiple events. It accounts for the overlap between events to avoid double-counting their intersection. The formula to find \(P(A \cup B)\) is:

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

Here, adding \(P(A)\) and \(P(B)\) includes the intersection \(P(A \cap B)\) twice, hence we subtract it once. For events A and B with probabilities 0.3 and 0.2, and an intersection of 0.1, the probability that at least one of them occurs is:

\(P(A \cup B) = 0.3 + 0.2 - 0.1 = 0.4\).

This principle avoids overestimating the likelihood that either one or both of the events will happen, a common trap when dealing with intersecting events.

De Morgan's Laws

De Morgan's Laws provide ways to express the complement of intersections and unions of sets in terms of each other. These laws are useful in simplifying complex probability problems by using the complements of events.

• The first law expresses the complement of the union: \((A \cup B)' = A' \cap B'\)
• The second law handles the complement of the intersection: \((A \cap B)' = A' \cup B'\)

Using De Morgan's Laws, we can find probabilities like \(P(A' \cup B)\) by transforming it with known probabilities. For instance, the calculation shown:
\(P(A' \cup B) = P((A \cap B)') = 1 - 0.2 = 0.8\), involves understanding how these events relate through their complements.

Intersection of Events

The intersection of events, denoted as \(A \cap B\), represents the probability that both events A and B occur simultaneously. It is crucial in probability theory to understand interactions between events. To find \(P(A \cap B)\), direct data often provides it; however, it can also be inferred through other probability statements.

Additionally, the complement of an intersection, such as \(A \cap B'\), is calculated by subtracting the joint occurrence \(P(A \cap B)\) from one event: \(P(A) - P(A \cap B)\). This gives us insights into scenarios where one event occurs but not the other. For example, \(P(A \cap B') = 0.3 - 0.1 = 0.2\).
Understanding the intersection allows us to navigate complex probabilities and helps illustrate how events can relate or intersect.