Problem 47 - Algebra and Trigonometry

Question

Find the probability of the indicated event if \(P(A)=0.25\) and \(P(B)=0.45\) $$P(A \cup B) \text { if } A, B \text { are mutually exclusive }$$

Step-by-Step Solution

Answer

The probability of \(A \cup B\) is 0.70.

Step 1 - Understand Mutually Exclusive Events

When two events are mutually exclusive, they cannot occur at the same time. This means that the intersection of events A and B is zero: \[ P(A \bigcap B) = 0 \]

Step 2 - Use the Probability Formula for Union of Two Events

The formula for the probability of the union of two events is: \[ P(A \bigcup B) = P(A) + P(B) - P(A \bigcap B) \]

Step 3 - Substitute Values into the Formula

Given that \(P(A) = 0.25\), \(P(B) = 0.45\), and \(P(A \bigcap B) = 0\):\[ P(A \bigcup B) = 0.25 + 0.45 - 0 \]

Step 4 - Simplify the Expression

Simplify the equation to find the probability of the union of A and B: \[ P(A \bigcup B) = 0.25 + 0.45 = 0.70 \]

Key Concepts

Probability

Probability is a measure that describes the likelihood of an event occurring. It is represented by a number between 0 and 1, where 0 means the event will not occur, and 1 means the event is certain to occur. For example, if you flip a fair coin, the probability of getting heads is 0.5.
Probability can be expressed as a fraction, decimal, or percentage. To calculate probability, use the formula: \[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]Understanding probability helps predict outcomes and make informed decisions based on those predictions.

Union of Events

The union of two events, denoted as \( A \cup B \), represents the event that either A occurs, B occurs, or both occur. Imagine you have two sets of outcomes; the union includes all possible outcomes from both sets. The formula to calculate the probability of the union of two events is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]Here, \( P(A \cap B) \) represents the intersection, or the event that both A and B occur. If events A and B are mutually exclusive (they cannot happen at the same time), then \( P(A \cap B) = 0 \).
So, the formula simplifies to: \[ P(A \cup B) = P(A) + P(B) \]Knowing how to compute the union of events is crucial in understanding combined probabilities.

Disjoint Events

Disjoint events (or mutually exclusive events) are events that cannot happen at the same time. If event A happens, event B cannot happen and vice versa. An example of this is rolling a die; you cannot get both a 2 and a 5 in one roll.
When dealing with disjoint events, the probability of their intersection is zero: \[ P(A \cap B) = 0 \]This property significantly simplifies probability calculations because the formula for the union of two events, \( P(A \cup B) \), excludes the intersection term.
Knowing that events are disjoint can often make solving probability problems faster and simpler.

Basic Probability Formulas

Several basic probability formulas are essential for solving different types of probability problems. Here are some key formulas:

• Probability of a single event: \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)
• Probability of the union of two events: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
• Probability of the complement of an event: \( P(A') = 1 - P(A) \)
• For mutually exclusive (disjoint) events: \( P(A \cup B) = P(A) + P(B) \)

These formulas form the foundation of probability theory and are incredibly useful when approaching a wide range of problems.