- Identify the individual probabilities from the table: \(P(\text{Thriller}) = 0.14\) and \(P(\text{Comedy}) = 0.28\).
- Apply the addition rule for mutually exclusive events: \(P(\text{Thriller or Comedy}) = P(\text{Thriller}) + P(\text{Comedy})\).
- Calculate the sum: \(0.14 + 0.28 = 0.42\).
- Express the result as a percentage: \(0.42 = \boxed{42\%}\).
Explanation
- Analyze the probability distribution and problem data
We are given the probability distribution of Mariah's movie collection, which consists of five mutually exclusive categories: Thriller, Drama, Comedy, Action, and Science fiction. Each category has an associated probability of a randomly chosen movie belonging to it. The probabilities are as follows:
- \(P(\text{Thriller}) = 0.14\)
- \(P(\text{Drama}) = 0.26\)
- \(P(\text{Comedy}) = 0.28\)
- \(P(\text{Action}) = 0.12\)
- \(P(\text{Science fiction}) = 0.2\)
The sum of these probabilities is \(0.14 + 0.26 + 0.28 + 0.12 + 0.2 = 1.0\), confirming that they form a complete probability distribution. Our objective is to determine the probability that a randomly selected movie is either a Thriller or a Comedy.
- Understand the application of the addition rule
Because the categories are mutually exclusive, a single movie cannot belong to both the Thriller and Comedy categories simultaneously. Therefore, the probability that a movie is either a Thriller or a Comedy is simply the sum of their individual probabilities. This is defined by the addition rule for mutually exclusive events:
\[P(\text{Thriller or Comedy}) = P(\text{Thriller}) + P(\text{Comedy})\]
- Calculate the combined probability
Now we perform the addition using the provided values:
\[P(\text{Thriller or Comedy}) = 0.14 + 0.28\]
The result of this addition is:
\[P(\text{Thriller or Comedy}) = 0.42\]
- Convert to percentage and conclude
To express this probability as a percentage, we multiply the decimal result by 100:
\[0.42 \times 100 = 42\%\]
Comparing this result to the given options, we identify that the correct choice is \(42\%\). Thus, the probability that a movie chosen at random is either a thriller or a comedy is \(\boxed{42\%}\).
Examples
In real life, this type of calculation is useful for inventory management in a library or a store. For example, if a bookstore manager knows the probability that a customer will purchase a specific genre of book, they can use this information to determine the likelihood of a customer choosing one of several popular genres, allowing them to optimize their stock levels accordingly. If \(P(\text{Mystery}) = 0.25\) and \(P(\text{Romance}) = 0.35\), the manager knows that the probability of a customer picking either genre is \(0.25 + 0.35 = 0.6\), or \(60\%\).