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 (1)
Use the conditional probability formula: P ( A ∣ B ) = P ( B ) P ( A ∩ B ) .
Identify P ( more than one dog and flea infestation ) = 0.032 and P ( flea infestation ) = 0.069 from the table.
Calculate the conditional probability: 0.069 0.032 ≈ 0.463768 .
Round the result to the nearest thousandths place: 0.464 .
Explanation
Understand the problem and provided data We are given a two-way relative frequency table that shows the relationship between dog ownership and flea infestations. We want to find the conditional relative frequency of having more than one dog, given there is a flea infestation. This is written as P(more than one dog | flea infestation).
Apply the conditional probability formula The formula for conditional probability is: P ( A ∣ B ) = P ( B ) P ( A ∩ B ) In our case, A is 'more than one dog' and B is 'flea infestation'. So we have: P ( more than one dog ∣ flea infestation ) = P ( flea infestation ) P ( more than one dog and flea infestation )
Identify the required probabilities from the table From the table, we can find the following probabilities: P ( more than one dog and flea infestation ) = 0.032 P ( flea infestation ) = 0.069
Calculate the conditional probability Now, we can plug these values into the conditional probability formula: P ( more than one dog ∣ flea infestation ) = 0.069 0.032 ≈ 0.4637681159
Round the result and conclude Rounding to the nearest thousandths place, we get 0.464.
Examples
Understanding conditional probabilities can help in various real-life scenarios. For instance, in medical research, it can be used to determine the probability of a patient having a disease given a positive test result. In marketing, it can help determine the probability of a customer buying a product given they have viewed an advertisement. In this case, understanding the conditional probability helps researchers understand the relationship between dog ownership and flea infestations.
