Question

Suppose that \(P(A)=.3\) and \(P(B)=.5 .\) If events \(A\) and \(B\) are mutually exclusive, find these probabilities: a. \(P(A \cap B)\) b. \(P(A \cup B)\)

Short answer

Answer: a) \(P(A \cap B) = 0\); b) \(P(A \cup B) = 0.8\)

Step-by-step solution

Find the probability of the intersection

Since events \(A\) and \(B\) are mutually exclusive, it means that they cannot occur at the same time. Therefore, the probability of their intersection will be zero. The probability of \(P(A \cap B) = 0\)

Find the probability of the union

For mutually exclusive events, the probability of their union is the sum of their individual probabilities: \(P(A \cup B) = P(A) + P(B)\) We know that \(P(A) = 0.3\) and \(P(B) = 0.5\), so we can substitute these values into the equation: \(P(A \cup B) = 0.3 + 0.5 = 0.8\) So, the probability of \(P(A \cup B) = 0.8\) The probabilities we found are: a. \(P(A \cap B) = 0\) b. \(P(A \cup B) = 0.8\)

Key concepts

P(A ∩ B) Probability

Understanding the probability of the intersection of two events, denoted as \(P(A \cap B)\), is a fundamental aspect of probability theory. It is the likelihood that both events A and B occur at the same time. However, things get straightforward when dealing with mutually exclusive events, where the occurrence of one event prevents the occurrence of the other.

As the exercise states, since events A and B cannot happen simultaneously—they are exclusive—the probability of their intersection is naturally zero. This concept is essential to master as it simplifies calculations and is a core principle in probability theory for distinct events. Symbolically, we represent this as \(P(A \cap B) = 0\).

When approaching problems of this nature, always remember to first assess the relationship between the events. The distinct nature of mutually exclusive events means there's no overlap, which is why the intersection probability is zero. It's a simple, yet critical point for correctly calculating probabilities in such scenarios.

P(A ∪ B) Probability

Now let's explore the probability of the union of two events, designated as \(P(A \cup B)\). In probability, the union refers to the likelihood that at least one of the events will occur. In mathematical terms, this figure encapsulates the possibilities of either event A happening, or event B happening, or both events occurring together.

However, our exercise revolves around mutually exclusive events, which means the events cannot happen at the same time. For such events, the calculation of the union probability is a breeze. All we need to do is to add the individual probabilities of each event, without the need to subtract the intersection (which is non-existent in this case). The resulting formula is thus \(P(A \cup B) = P(A) + P(B)\).

This approach is vitally important not only for its simplicity but also for its frequent application in various probability problems. It provides a way to swiftly calculate the overall likelihood of several distinct outcomes occurring.

Probability Theory Basics

Diving into the basics of probability theory is like exploring the language of uncertainty. At its most foundational level, this branch of mathematics deals with quantifying the likelihood of events within a given set of possibilities.

The core of probability theory includes crucial concepts such as events, sample spaces, and probabilities themselves. An event represents a possible outcome that we are focusing on, and the sample space encompasses all conceivable outcomes. The probability of any event is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

Essentially, there are a few key points to grasp:

• Probability is always a non-negative value.
• The sum of probabilities for all possible outcomes in a sample space is 1.
• The calculation methods can vary whether events are independent, mutually exclusive, or neither.

When starting with probability, remember these basic principles since they underpin more complex concepts and calculations. With a solid understanding of these fundamentals, students are well-equipped to tackle a wide array of probabilistic scenarios.