Problem 11 - Basic Statistics for Business and Economics with Student

Question

The events \(A\) and \(B\) are mutually exclusive. Suppose \(P(A)=.30\) and \(P(B)=.20 .\) What is the probability of either \(A\) or \(B\) occurring? What is the probability that neither \(A\) nor \(B\) will happen?

Step-by-Step Solution

Answer

The probability of either \(A\) or \(B\) occurring is 0.50, and neither occurring is also 0.50.

Step 1: Understanding Mutually Exclusive Events

If two events are mutually exclusive, it means that they cannot happen at the same time. In probability terms, this implies that the intersection of these events is zero, i.e., \( P(A \cap B) = 0 \).

Step 2: Calculating Probability of Either Event Occurring

Since events \(A\) and \(B\) are mutually exclusive, the probability of either \(A\) or \(B\) occurring is the sum of their individual probabilities. Thus, \[ P(A \cup B) = P(A) + P(B) = 0.30 + 0.20 = 0.50. \]

Step 3: Calculating the Probability of Neither Event Occurring

The probability that neither \(A\) nor \(B\) occurs is the complement of the probability that \(A\) or \(B\) occurs. This means \[ P(\text{neither } A \text{ nor } B) = 1 - P(A \cup B) = 1 - 0.50 = 0.50. \]

Key Concepts

Probability Calculation

Calculating probabilities is a fundamental part of understanding how likely an event is to occur. When we talk about probability calculation, it involves finding the numerical likelihood of an event happening. This is typically expressed as a number between 0 and 1.
For mutually exclusive events, this calculation becomes a bit more specific. **Mutually exclusive events** mean that if one event happens, the other cannot. So in terms of probability, like in our example with events \(A\) and \(B\), the formula simplifies to adding the individual probabilities together when you want to find the likelihood of either event occurring:

• The formula is: \( P(A \cup B) = P(A) + P(B) \).
• In our example, \( P(A) = 0.30 \) and \( P(B) = 0.20 \).
• Therefore, \( P(A \cup B) = 0.30 + 0.20 = 0.50 \).

This means there's a 50% chance that either event \(A\) or event \(B\) will occur. Calculating these probabilities helps in predicting outcomes and making informed decisions.

Complement Rule

The complement rule is a handy tool in probability that allows us to easily find the likelihood of an event not happening. Simply put, the complement of a probability is what is left over when you subtract the probability of the event from 1.
In our context, we want to find the probability that neither event \(A\) nor event \(B\) will occur. These two events together have a total probability of 0.50 of happening, as they are mutually exclusive.

• The complement rule formula we use here is \( P(\text{neither } A \text{ nor } B) = 1 - P(A \cup B) \).
• Thus, \( P(\text{neither } A \text{ nor } B) = 1 - 0.50 = 0.50 \).

So, there's also a 50% chance that neither event \(A\) nor \(B\) will occur. By using the complement rule, we have an easy way to see the other side of probability — what doesn't happen.

Probability Theory

Probability theory is the branch of mathematics that deals with the study of random events. It provides the foundation for understanding and managing uncertainty.
When we consider mutually exclusive events, probability theory simplifies how we handle these situations. With mutually exclusive events like \(A\) and \(B\), it is impossible for them to occur simultaneously, meaning:

• Events \(A\) and \(B\) are separate paths, and only one can be followed in a single trial.
• Their intersection is zero: \( P(A \cap B) = 0 \).
• This characteristic allows us to add their probabilities directly to find the union \( P(A \cup B) \).

Probability theory not only helps us calculate specific outcomes but also equips us with tools like the complement rule to understand all possible scenarios, ensuring we account for what happens or doesn't, providing a complete picture of potential outcomes.