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A pet store asked customers what type of pet they own. The results of the survey are organized in the table below:

| | Cat | Dog | Hamster | Total |
|--------|-----|-----|---------|-------|
| Male | 11 | 19 | 7 | 37 |
| Female | 8 | 14 | 4 | 26 |
| Total | 19 | 33 | 11 | 63 |

If [tex]\( C \)[/tex] is the event of owning a cat and [tex]\( F \)[/tex] is the event that a pet is female, find [tex]\( P(C \cap F) \)[/tex]. Round your answer to two decimal places.



Answer :

GinnyAnswer
Let's solve the problem step-by-step to find [tex]\( P(C \cap F) \)[/tex], the probability that a pet owner is both female and owns a cat.

1. Identify the Total Number of Pet Owners:
The total number of pet owners surveyed is given in the table as 63.

2. Identify the Number of Female Cat Owners:
According to the table, there are 8 female cat owners.

3. Calculate [tex]\( P(C \cap F) \)[/tex]:
[tex]\( P(C \cap F) \)[/tex] is the probability that a randomly selected pet owner is both female and owns a cat. This can be calculated using the formula:
[tex]\[ P(C \cap F) = \frac{\text{Number of Female Cat Owners}}{\text{Total Number of Pet Owners}} \][/tex]
Plugging in the numbers from the table:
[tex]\[ P(C \cap F) = \frac{8}{63} \][/tex]

4. Simplify and Round the Probability:
We need to round the probability to two decimal places.

The fractional value [tex]\( \frac{8}{63} \)[/tex] is approximately 0.12698412698412698 when expressed as a decimal.

5. Round to Two Decimal Places:
Rounding 0.12698412698412698 to two decimal places gives us 0.13.

So, the probability [tex]\( P(C \cap F) \)[/tex] is [tex]\( 0.13 \)[/tex].

Answer:

To find the probability of owning a female cat, we need to find the number of females who have a cat and divide by the total number of female respondents:

P(C ∩ F) = (Number of females who own a cat)/Total number of females

P(C ∩ F) = (8)/(26)

P(C ∩ F) ≈ 0.308

So, the probability of owning a female cat is approximately 0.308.

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