Problem 52 - Algebra and Trigonometry

Question

According to the American Pet Products Manufacturers Association's \(2017-2018\) National Pet Owners Survey, there is a \(68 \%\) probability that a U.S. household owns a pet. If a U.S. household is randomly selected, what is the probability that it does not own a pet?

Step-by-Step Solution

Answer

The probability that a U.S. household does not own a pet is 0.32.

Step 1: Understand the given probability

The problem states that there is a 68% probability that a U.S. household owns a pet. This can be written as a decimal: \[ P(\text{Owns a pet}) = 0.68 \]

Step 2: Understand the complementary probability

The probability that a household does not own a pet is the complement of the probability that it does own a pet. The sum of these two probabilities must be 1 since either a household owns a pet or it does not.

Step 3: Calculate the complementary probability

To find the probability that a household does not own a pet, subtract the given probability from 1: \[ P(\text{Does not own a pet}) = 1 - P(\text{Owns a pet}) = 1 - 0.68 \]

Step 4: Simplify the expression

Perform the subtraction to find the probability that a household does not own a pet: \[ P(\text{Does not own a pet}) = 1 - 0.68 = 0.32 \]

Key Concepts

probability

Probability is a measure of how likely an event is to occur. In mathematics, it is often represented as a number between 0 and 1. A probability of 0 means an event will not happen, while a probability of 1 means it will definitely happen. For example, if the probability of rain tomorrow is 0.8, it suggests an 80% chance of rain.
In our exercise, we start with a given probability that a U.S. household owns a pet, which is 68%. We can express this probability as a decimal by dividing by 100: \( P(\text{Owns a pet}) = 0.68 \). This step is crucial because it allows us to use this probability in our calculations for further steps.

complementary events

Complementary events are two outcomes of an event that cover all possible outcomes together. One event is the complement of the other. The sum of the probabilities of complementary events is always 1. For instance, if the probability of event A happening is 0.7, the probability of event A not happening is 0.3, because 0.7 + 0.3 = 1.
In our particular problem, the events we are considering are whether a household owns a pet or does not own a pet. Since these are complementary events, we use the given probability to find the complement. If \( P(\text{Owns a pet}) = 0.68 \), then the complement is \( P(\text{Does not own a pet}) = 1 - 0.68 = 0.32 \). This simple subtraction gives us the probability of the complementary event.

calculation steps

In order to solve the problem, follow these calculation steps:
1. **Understand the given probability:** We know that the probability of a household owning a pet is 68%, so we write it as a decimal: \( P(\text{Owns a pet}) = 0.68 \).
2. **Understand complementary probability:** Recognize that the probability of a household not owning a pet is the complement of the probability of it owning a pet. The sum of both probabilities must equal 1.
3. **Calculate the complementary probability:** Subtract the given probability from 1 to find the probability that a household does not own a pet. This is done by \( 1 - 0.68 \).
4. **Simplify the expression:** Perform the subtraction: \( P(\text{Does not own a pet}) = 1 - 0.68 = 0.32 \).
By following these steps, you can reliably find complementary probabilities for other problems too.