Every family in Happyland has either has a car or a motor scooter or both. 75% of the families have a car. 81% of the families have a scooter. A family is selected at random and it is found that they have a car. Find the probability they also have a scooter.

Give your answer as a fraction.

Let's define the events:

- \( C \): The event that a family has a car.
- \( S \): The event that a family has a scooter.

We are given:
- \( P(C) = 0.75 \) (75% of the families have a car)
- \( P(S) = 0.81 \) (81% of the families have a scooter)
- We need to find the probability that a family also has a scooter, given that they already have a car. This is \( P(S \mid C) \), the conditional probability.

The conditional probability formula is:
\[
P(S \mid C) = \frac{P(S \cap C)}{P(C)}
\]
Where:
- \( P(S \cap C) \) is the probability that a family has both a car and a scooter.

Using the inclusion-exclusion principle, we can find \( P(S \cap C) \):
\[
P(S \cup C) = P(S) + P(C) - P(S \cap C)
\]
Since every family has either a car or a scooter or both, \( P(S \cup C) = 1 \). Thus, we have:
\[
1 = P(S) + P(C) - P(S \cap C)
\]
Substitute the known values of \( P(S) \) and \( P(C) \):
\[
1 = 0.81 + 0.75 - P(S \cap C)
\]
\[
P(S \cap C) = 0.81 + 0.75 - 1 = 0.56
\]
Now, we can calculate \( P(S \mid C) \):
\[
P(S \mid C) = \frac{P(S \cap C)}{P(C)} = \frac{0.56}{0.75} = \frac{56}{75}
\]
Thus, the probability that a family has a scooter given that they have a car is \( \frac{56}{75} \).

Would you like more details on this? Here are 5 related questions:

1. What is the probability a family has both a car and a scooter?
2. What is the probability a family has only a car and not a scooter?
3. What is the probability a family has neither a car nor a scooter?
4. Can you explain conditional probability using another example?
5. How would this problem change if 100% of families had either a car or a scooter but not both?

**Tip:** Conditional probability helps refine probabilities based on additional known information, making it useful in decision-making and predictions.

Every family in Happyland has either a car or a motor scooter or both. 75% of the families have a car. 81% of the families have a scooter. A family is selected at random and it is found that they have a car. Find the probability they also have a scooter.

Understand how to calculate the probability that a family in Happyland has both a car and a scooter, given they have a car. This problem involves key concepts in probability, including conditional probability and the inclusion-exclusion principle. Suitable for students in Grades 10-12.

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