Question

Suppose the probability that a U.S. resident has traveled to Canada is \(0.18\), to Mexico is \(0.09\), and to both countries is 0.04. What's the probability that an American chosen at random has a) traveled to Canada but not Mexico? b) traveled to either Canada or Mexico? c) not traveled to either country?

Short answer

a) 0.14 b) 0.23 c) 0.77

Step-by-step solution

Understand the problem

We are given probabilities related to U.S. residents traveling to Canada, Mexico, and both countries. We need to determine the probabilities for specific scenarios involving travel to these countries.

Define the formula for Part A

To find the probability that a resident has traveled to Canada but not Mexico, we use the formula: \( P(C \text{ and not } M) = P(C) - P(C \text{ and } M) \). We are given \( P(C) = 0.18 \) and \( P(C \text{ and } M) = 0.04 \).

Calculate for Part A

Substitute the values into the formula: \( P(C \text{ and not } M) = 0.18 - 0.04 = 0.14 \). This is the probability of a resident traveling to Canada but not to Mexico.

Define the formula for Part B

For residents traveling to either Canada or Mexico, use the formula: \( P(C \text{ or } M) = P(C) + P(M) - P(C \text{ and } M) \). Substitute the probabilities given: \( P(C) = 0.18 \), \( P(M) = 0.09 \), \( P(C \text{ and } M) = 0.04 \).

Calculate for Part B

Plug the values into the formula: \( P(C \text{ or } M) = 0.18 + 0.09 - 0.04 = 0.23 \). This is the probability that a resident has traveled to either Canada or Mexico.

Define the formula for Part C

For residents who have not traveled to either country, use \( P(\text{not either}) = 1 - P(C \text{ or } M) \).

Calculate for Part C

Use the result from Step 5: \( P(\text{not either}) = 1 - 0.23 = 0.77 \). This gives us the probability that a resident has not traveled to either Canada or Mexico.

Key concepts

Conditional Probability

Conditional probability helps us understand the likelihood of an event happening given that another event has already taken place. It is typically written as \( P(A | B) \), which means the probability of event \( A \) occurring given that event \( B \) has occurred. Consider this situation: if you're trying to find the probability of a U.S. resident traveling to Canada and then not traveling to Mexico, you're dealing with a conditional probability.
To calculate this, you would use the relationship:
\[ P(C \text{ and not } M) = P(C) - P(C \text{ and } M) \]
In this context, you first find the probability of traveling to Canada and then subtract the overlap of those who traveled to both Canada and Mexico. This method isolates the probability of traveling only to Canada. By refining outcomes based on prior events, conditional probability serves as a key tool for making accurate predictions in real-world scenarios.

Independent Events

In probability, events are considered independent if the occurrence of one doesn't affect the occurrence of another. In other words, two events \( A \) and \( B \) are independent if \( P(A \cap B) = P(A) \cdot P(B) \).
However, in this exercise, traveling to Canada and Mexico are not independent events because the probability of residents traveling to both does not equal the product of their individual probabilities.
The given probabilities are:

• \( P(C) = 0.18 \)
• \( P(M) = 0.09 \)
• \( P(C \cap M) = 0.04 \)

To check for independence, we would use \( P(C) \cdot P(M) \) and compare it with \( P(C \cap M) \). If \( 0.18 \times 0.09 \) does not equal \( 0.04 \) (which it doesn't as the calculated product is 0.0162), it confirms that the events are not independent. Understanding whether events are independent helps in simplifying complex probability calculations, especially when evaluating multiple possibilities.

Set Operations in Probability

Set operations in probability allow us to handle events using the language of sets. Key operations include union, intersection, and complement. These operations help in determining probabilities of combined or opposite events.
For example, the union operation, denoted as \( A \cup B \), refers to either event \( A \) or event \( B \) occurring. This is represented by the formula:
\[ P(C \cup M) = P(C) + P(M) - P(C \cap M) \]
Here, by adding the probabilities of traveling to Canada and Mexico and subtracting the overlap (those who went to both), you find the probability of residents traveling to either country.
The concept of complement comes into play when looking for \( P(\text{not either}) \), which means neither event occurs:
\[ P(\text{not either}) = 1 - P(C \cup M) \]
This operation efficiently calculates the opposite scenario, providing a way to evaluate entire probability spaces. By integrating set operations, we gain a structured approach to handle complex probability scenarios.