Problem 68 - Basic Statistics for Business and Economics with Student

Question

IIt was found that 60 percent of the tourists to China visited the Forbidden City, the Temple of Heaven, the Great Wall, and other historical sites in or near Beijing. Forty percent visited Xi'an with its magnificent terracotta soldiers, horses, and chariots, which lay buried for over 2,000 years. Thirty percent of the tourists went to both Beijing and Xi'an. What is the probability that a tourist visited at least one of these places?

Step-by-Step Solution

Answer

70% probability a tourist visited at least one of the places.

Step 1: Define Known Probabilities

Identify the given probabilities for visiting each site. Let \( A \) be the event a tourist visited Beijing (including the Forbidden City and other sites) and \( B \) be the event a tourist visited Xi'an. It is given that \( P(A) = 0.60 \), \( P(B) = 0.40 \), and \( P(A \cap B) = 0.30 \).

Step 2: Use the Union Formula

To find the probability that a tourist visited at least one of Beijing or Xi'an, we need to calculate \( P(A \cup B) \). We use the formula for the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \].

Step 3: Plug in Known Values

Substitute the known probabilities into the union formula: \[ P(A \cup B) = 0.60 + 0.40 - 0.30 \].

Step 4: Calculate the Result

Perform the arithmetic to get \( P(A \cup B) = 1.00 - 0.30 = 0.70 \). Thus, the probability that a tourist visited at least one of these places is 0.70.

Key Concepts

Union of Events

In probability theory, the union of events refers to a scenario where at least one of the given events occurs. For instance, if we have two events, like visiting Beijing or Xi'an, the union is the chance of visiting at least one of these places. The union is represented by the mathematical symbol \( \cup \), which means "or" in this context. The formula to calculate the probability of the union of two events, say \( A \) and \( B \), is \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). This formula subtracts the overlap because adding \( P(A) \) and \( P(B) \) counts it twice. This concept helps us understand overlapping probabilities by ensuring we don't "double count" the tourists visiting both Beijing and Xi'an.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already happened. Although not directly heavy-weighted in this exercise, it supports our understanding of probability relationships.Imagine two interconnected events: visiting Beijing and Xi'an. Knowing one has occurred can change the likelihood of the other. This is mathematically represented as \( P(A | B) \), meaning the probability of \( A \) occurring given that \( B \) occurred.Understanding this concept helps uncover dependencies between events. Even if we don't utilize it directly, recognizing how events might overlap or influence one another can clarify union calculations.

Arithmetic Calculations

Arithmetic calculations are crucial for resolving probability exercises. In our example, we used basic arithmetic operations to find the probability of a tourist visiting at least one of the places.Once we identified known probabilities: 0.60 for Beijing, 0.40 for Xi'an, and 0.30 for both, we employed the arithmetic inside the union formula. By substituting, \( 0.60 + 0.40 - 0.30 \), we achieved the final result of 0.70.Arithmetic is not just about numbers; it involves careful handling of data to reach logical conclusions. These simple calculations underscore the importance of arithmetic when addressing probability issues.