Answered

Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

Calculating and Comparing Probabilities

A veterinarian surveys her clients and finds that 32 percent of the households have dogs, 25 percent have cats, and 11 percent have both dogs and cats. Let event [tex]C[/tex] be choosing a client who has cats and let event [tex]D[/tex] be choosing a client who has dogs. Which statements are true? Check all that apply.

- [tex]P(C \mid D)=0.78[/tex]
- [tex]P(D \mid C)=0.44[/tex]
- [tex]P(C \cap D)=0.11[/tex]
- [tex]P(C \cap D)=P(D \cap C)[/tex]
- [tex]P(C \mid D)=P(D \mid C)[/tex]

Sagot :

GinnyAnswer
Let's analyze the given information and calculate the necessary probabilities to determine which statements are true.

Given:
- [tex]\( P(C) = 0.25 \)[/tex] (Probability of having cats)
- [tex]\( P(D) = 0.32 \)[/tex] (Probability of having dogs)
- [tex]\( P(C \cap D) = 0.11 \)[/tex] (Probability of having both cats and dogs)

We are to evaluate the following statements:
1. [tex]\( P(C \mid D) = 0.78 \)[/tex]
2. [tex]\( P(D \mid C) = 0.44 \)[/tex]
3. [tex]\( P(C \cap D) = 0.11 \)[/tex]
4. [tex]\( P(C \cap D) = P(D \cap C) \)[/tex]
5. [tex]\( P(C \mid D) = P(D \mid C) \)[/tex]

### Step-by-Step Calculations:

1. Calculate [tex]\( P(C \mid D) \)[/tex]:

[tex]\( P(C \mid D) = \frac{P(C \cap D)}{P(D)} \)[/tex]

Substituting the given values:

[tex]\( P(C \mid D) = \frac{0.11}{0.32} \approx 0.34375 \)[/tex]

This means [tex]\( P(C \mid D) \approx 0.34375 \)[/tex], not 0.78. Hence, the statement [tex]\( P(C \mid D) = 0.78 \)[/tex] is false.

2. Calculate [tex]\( P(D \mid C) \)[/tex]:

[tex]\( P(D \mid C) = \frac{P(C \cap D)}{P(C)} \)[/tex]

Substituting the given values:

[tex]\( P(D \mid C) = \frac{0.11}{0.25} = 0.44 \)[/tex]

This means [tex]\( P(D \mid C) = 0.44 \)[/tex]. Hence, the statement [tex]\( P(D \mid C) = 0.44 \)[/tex] is true.

3. Verify [tex]\( P(C \cap D) = 0.11 \)[/tex]:

[tex]\( P(C \cap D) = 0.11 \)[/tex]

This verifies that the given probability [tex]\( P(C \cap D) = 0.11 \)[/tex] is correct. Hence, the statement [tex]\( P(C \cap D) = 0.11 \)[/tex] is true.

4. Verify [tex]\( P(C \cap D) = P(D \cap C) \)[/tex]:

By the definition of intersection in probability, [tex]\( P(C \cap D) \)[/tex] is the same as [tex]\( P(D \cap C) \)[/tex].

Hence, the statement [tex]\( P(C \cap D) = P(D \cap C) \)[/tex] is true.

5. Verify [tex]\( P(C \mid D) = P(D \mid C) \)[/tex]:

From the calculations:
- [tex]\( P(C \mid D) \approx 0.34375 \)[/tex]
- [tex]\( P(D \mid C) = 0.44 \)[/tex]

This shows [tex]\( P(C \mid D) \neq P(D \mid C) \)[/tex]. Hence, the statement [tex]\( P(C \mid D) = P(D \mid C) \)[/tex] is false.

### Summary of Statements:

- [tex]\( P(C \mid D) = 0.78 \)[/tex] is false.
- [tex]\( P(D \mid C) = 0.44 \)[/tex] is true.
- [tex]\( P(C \cap D) = 0.11 \)[/tex] is true.
- [tex]\( P(C \cap D) = P(D \cap C) \)[/tex] is true.
- [tex]\( P(C \mid D) = P(D \mid C) \)[/tex] is false.

Thus, the true statements are:
- [tex]\( P(D \mid C) = 0.44 \)[/tex]
- [tex]\( P(C \cap D) = 0.11 \)[/tex]
- [tex]\( P(C \cap D) = P(D \cap C) \)[/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.