The probability that Clara is allowed to use the car to go out tonight is 60%. The probability that Clara can use the car and there’s no gas in it is 30%. What is the probability that there’s no gas in the car given that Clara can’t use the car?

Let's break this problem down using conditional probability.

Let:
- \( P(A) \) = Probability that Clara can use the car = 0.60.
- \( P(A') \) = Probability that Clara can't use the car = 1 - \( P(A) = 0.40 \).
- \( P(B|A) \) = Probability that there’s no gas in the car given that Clara can use the car. This probability is part of the information provided as \( P(A \cap B) = 0.30 \), which means the joint probability that Clara can use the car and there's no gas in it is 0.30.

We are asked to find \( P(B|A') \), the probability that there’s no gas in the car given that Clara **can’t** use the car. This can be approached using Bayes' Theorem or direct reasoning.

First, let's calculate the total probability that there's no gas in the car, \( P(B) \).

### Step 1: Find \( P(B) \) (the total probability of no gas)
We can use the law of total probability:
\[
P(B) = P(B|A)P(A) + P(B|A')P(A')
\]
We already know \( P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.30}{0.60} = 0.50 \). So,

\[
P(B) = (0.50)(0.60) + P(B|A')(0.40)
\]
\[
P(B) = 0.30 + P(B|A')(0.40)
\]

### Step 2: Solve for \( P(B|A') \)
Now, rearrange the equation to solve for \( P(B|A') \):
\[
P(B) - 0.30 = P(B|A')(0.40)
\]
Thus, the unknown \( P(B|A') \) can be found if we calculate the total probability \( P(B) \). However, we still need more information to completely determine it, unless some reasonable assumptions or additional context is provided.

Would you like me to go deeper into possible assumptions or clarify further? Let me know if you have more details!

Explore how to calculate the conditional probability that there's no gas in the car given Clara can't use the car. This problem involves using the law of total probability and conditional probability formulas, with detailed explanations for students in Grades 10-12.

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