Problem 53 - Algebra and Trigonometry

Question

According to the American Pet Products Manufacturers Association's \(2017-2018\) National Pet Owners Survey, there is a \(38 \%\) probability that a U.S. household owns a cat. If a U.S. household is randomly selected, what is the probability that it does not own a cat?

Step-by-Step Solution

Answer

The probability that a U.S. household does not own a cat is 0.62.

Step 1 - Understand the Given Probability

The problem states that there is a 38% probability that a U.S. household owns a cat. This can be written as a probability of 0.38.

Step 2 - Recall the Complement Rule

The probability that an event does not occur is equal to 1 minus the probability that the event does occur. This is called the complement rule.

Step 3 - Apply the Complement Rule

To find the probability that a U.S. household does not own a cat, subtract the probability that it does own a cat from 1. Mathematically: \[ P(\text{not owning a cat}) = 1 - P(\text{owning a cat}) \]

Step 4 - Calculate the Probability

Substitute the given probability into the complement formula: \[ P(\text{not owning a cat}) = 1 - 0.38 = 0.62 \]

Key Concepts

probability

Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means the event will not happen at all, and 1 means the event is certain to happen. In our exercise, we are dealing with the probability of households owning or not owning a cat. The probability of a household owning a cat is given as 38%, or 0.38 in decimal form. Understanding probability helps us in predicting outcomes and making informed decisions.

complementary events

Complementary events are pairs of events that encompass all possible outcomes for a situation. For any event, the complement is the event that the original event does not happen. For example, if our event is 'owning a cat,' the complementary event is 'not owning a cat.' The sum of the probabilities of an event and its complement is always 1. This is a key concept in probability because it allows us to calculate the probability of the complementary event by subtracting the probability of the event from 1. In our case, since there is a 0.38 probability that a household owns a cat, the probability that it does not own a cat is 1 - 0.38.

probability calculation

The process of calculating probabilities usually involves using simple mathematical operations. In our exercise, we applied the complement rule to find the probability of the complementary event. Here are the steps to perform the calculation:
- Start with the given probability that a household owns a cat: 0.38.
- Apply the complement rule, which is expressed mathematically as: \[ P(\text{not owning a cat}) = 1 - P(\text{owning a cat}) \]
- Substitute the given probability into the formula: \[ P(\text{not owning a cat}) = 1 - 0.38 = 0.62 \]
This means there is a 0.62 or 62% probability that a randomly selected U.S. household does not own a cat. Probability calculations help us quantify and understand the likelihood of different outcomes in various scenarios.