Problem 55 - Algebra and Trigonometry

Question

According to a 2016 Gallup survey, \(26 \%\) of U.S. adults visited a casino within the past year. If a U.S. adult is selected at random, what is the probability that he or she has not visited a casino within the past year?

Step-by-Step Solution

Answer

The probability is 0.74.

Step 1 - Understand the probability given

The problem states that 26% of U.S. adults visited a casino within the past year. This percentage can be converted to its probability form by recognizing that 26% is equivalent to 0.26.

Step 2 - Set up the complement

To find the probability that a U.S. adult has not visited a casino within the past year, use the concept of complement. The complement rule states that the probability of an event not occurring is equal to 1 minus the probability of the event occurring.

Step 3 - Apply the complement rule

Calculate the probability that an adult has not visited a casino within the past year by subtracting the probability that an adult has visited a casino from 1: \[ P(\text{not visited}) = 1 - P(\text{visited}) \]Substitute the given probability into the equation:\[ P(\text{not visited}) = 1 - 0.26 \]\[ P(\text{not visited}) = 0.74 \]

Step 4 - State the result

The probability that a randomly selected U.S. adult has not visited a casino within the past year is 0.74.

Key Concepts

Complement Rule

In probability theory, the complement rule is a handy tool for solving problems like the one in the original exercise. The basic idea is simple: if you know the probability of an event happening, you can easily find the probability of it not happening.
Think of it this way: every possible outcome in a probability experiment sums up to 1. So, if you have the probability of an event, say event 'A', you can find its complement (the probability of event 'A' not happening) by subtracting the probability of event 'A' from 1.
This can be summarized as:
\[ P(A^c) = 1 - P(A) \] where \ P(A^c) \ is the complement of event 'A'.
Using this rule helps avoid mistakes in probability calculations and is especially useful when direct calculation of the non-event is tricky.

Event Probability

An event's probability refers to the likelihood that it will occur. It is a number between 0 and 1, where 0 means the event cannot happen and 1 means it definitely will.
For example, the original exercise states that 26% of U.S. adults visited a casino within the past year. This 26% can be converted into a probability by dividing it by 100, giving us 0.26.
When dealing with probabilities, remember to consider:

• If an event is certain to happen, its probability is 1.
• If an event is certain not to happen, its probability is 0.
• For other events, the probability will be a number between 0 and 1.

In the case of our exercise, knowing the probability of visiting a casino helps us use the complement rule to find the probability of not visiting.

Percent to Probability Conversion

Converting percentages to probabilities is a crucial skill in solving probability problems. Percentages are often used in surveys and reports because they are more intuitive to understand.
To convert a percentage to a probability, simply divide the percentage value by 100. For instance, if we know that 26% of people visited a casino, we can convert that to a probability as follows:
\[ \text{Probability} = \frac{\text{Percentage}}{100} \]
Applying this to our example: \[ P(\text{visited}) = \frac{26}{100} = 0.26 \]
This simple step is vital if we are to use the percentage information in further calculations, such as using the complement rule. Always remember, converting percentages to probabilities allows us to make precise mathematical calculations.