Some researchers claim there is an association between owning a dog and developing a flea infestation at home. They took a random sample of people living in St. Louis, Missouri, and recorded whether they had a dog and whether they had a flea infestation in their home. The results are shown in the following two-way relative frequency table:
| | Has Flea Infestation | Does Not Have Flea Infestation | Row Totals |
| :-------------------- | :------------------- | :----------------------------- | :--------- |
| Does not have a dog | 0.015 | 0.652 | 0.667 |
| Has one dog | 0.022 | 0.261 | 0.283 |
| Has more than one dog | 0.032 | 0.019 | 0.050 |
| Column Totals | 0.069 | 0.931 | 1 |
What is the conditional relative frequency of having more than one dog, given there is a flea infestation? Round your answer to the nearest thousandths place.
Answer (2)
Define events: A = More than one dog, B = Flea infestation.
Use conditional probability formula: P ( A ∣ B ) = P ( B ) P ( A ∩ B ) .
Extract probabilities from the table: P ( A ∩ B ) = 0.032 , P ( B ) = 0.069 .
Calculate and round: P ( A ∣ B ) = 0.069 0.032 ≈ 0.464 .
Explanation
Understand the problem We are asked to find the conditional relative frequency of having more than one dog, given there is a flea infestation. This is a conditional probability problem. Let's denote the event of having more than one dog as A, and the event of having a flea infestation as B. We need to find P(A|B), which is the probability of A given B.
State the formula for conditional probability The formula for conditional probability is: P ( A ∣ B ) = P ( B ) P ( A ∩ B ) Where:
P ( A ∩ B ) is the probability of both A and B occurring.
P ( B ) is the probability of B occurring.
Identify probabilities from the table From the table, we can find the following probabilities:
P ( A ∩ B ) = P(More than One Dog and Flea Infestation) = 0.032
P ( B ) = P(Flea Infestation) = 0.069
Calculate the conditional probability Now, we can calculate the conditional probability: P ( A ∣ B ) = 0.069 0.032 ≈ 0.46376811594
Round the result Rounding the result to the nearest thousandths place, we get 0.464.
State the final answer Therefore, the conditional relative frequency of having more than one dog, given there is a flea infestation, is approximately 0.464.
Examples
Conditional probability is used in many real-world scenarios. For example, in medical diagnosis, it helps determine the probability of a disease given certain symptoms. In marketing, it can predict the likelihood of a customer buying a product based on their past purchases. In finance, it can assess the risk of an investment based on market conditions. Understanding conditional probability allows us to make informed decisions based on available data.
The conditional relative frequency of having more than one dog, given there is a flea infestation, is calculated as approximately 0.464. This is determined by using the conditional probability formula, where we divide the probability of having more than one dog and a flea infestation by the total probability of having a flea infestation. Rounding to the nearest thousandths place gives us the final answer.
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