- The events 'Thriller' and 'Comedy' are mutually exclusive, meaning they cannot happen at the same time.
- The probability of picking one or the other is the sum of their individual probabilities: \(P(\text{Thriller}) + P(\text{Comedy}) = 0.14 + 0.28\).
- The total probability is \(0.42\).
- Expressed as a percentage, the probability is \(\boxed{42\%}\).
Explanation
- Analyze the data and identify the events
The problem asks us to determine the probability that a movie selected at random from Mariah's collection is either a 'Thriller' or a 'Comedy'. From the provided probability distribution table, we have the individual probabilities:
- The probability of choosing a 'Thriller' is \(P(\text{Thriller}) = 0.14\).
- The probability of choosing a 'Comedy' is \(P(\text{Comedy}) = 0.28\).
Each movie in the collection belongs to exactly one of the five categories, which means these events are mutually exclusive.
- Apply the addition rule for mutually exclusive events
Because the events 'Thriller' and 'Comedy' are mutually exclusive, the probability of selecting a movie that is either a thriller or a comedy is the sum of their individual probabilities. This is expressed by the addition rule:
\[P(\text{Thriller or Comedy}) = P(\text{Thriller}) + P(\text{Comedy})\]
Substituting the given values into the formula:
\[P(\text{Thriller or Comedy}) = 0.14 + 0.28 = 0.42\]
- Calculate the final probability and select the option
To convert the decimal result to a percentage, we multiply it by \(100\):
\[0.42 \times 100 = 42\%\]
Comparing this result with the provided options, we see that \(42\%\) is the correct choice.
Examples
Understanding how to combine probabilities of mutually exclusive events is fundamental in statistics and real-world decision-making. For example, if you are managing a warehouse inventory and know the probability of receiving different types of items, you can calculate the total probability of receiving a specific subset of items to better plan your storage and logistics. If the probability of a shipment containing electronics is \(0.15\) and the probability of it containing books is \(0.25\), the probability that it contains either electronics or books is \(0.15 + 0.25 = 0.40\), or \(40\%\).