At LaGuardia Airport for a certain nightly flight, the probability that it will rain is 0.18 and the probability that the flight will be delayed is 0.13. The probability that it will not rain and the flight will leave on time is 0.7. What is the probability that it is raining if the flight has been delayed? Round your answer to the nearest thousandth.


We are asked to find the probability that it is raining given that the flight has been delayed. This is a conditional probability problem, and we can use Bayes' theorem to solve it.

### Bayes' Theorem:
\[
P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)}
\]
Where:
- \( P(A \mid B) \) is the probability that it is raining given that the flight has been delayed.
- \( P(A) \) is the probability that it is raining.
- \( P(B) \) is the probability that the flight has been delayed.
- \( P(B \mid A) \) is the probability that the flight is delayed given that it is raining.

### Step 1: Identify given information
- \( P(\text{rain}) = 0.18 \)
- \( P(\text{delayed}) = 0.13 \)
- \( P(\text{not rain} \text{ and not delayed}) = 0.70 \)

We need to calculate \( P(\text{rain} \mid \text{delayed}) \).

### Step 2: Compute the probability of the flight being delayed, \( P(\text{delayed}) \)

To calculate this, we need to split the possible outcomes:
1. When it rains, the flight can be delayed or not delayed.
2. When it does not rain, the flight can also be delayed or not delayed.

Let’s break it down:

- \( P(\text{not rain and not delayed}) = 0.70 \), which means that the probability of it not raining and the flight leaving on time is 0.70.
- The probability of it not raining is \( P(\text{not rain}) = 1 - P(\text{rain}) = 1 - 0.18 = 0.82 \).

Therefore, the probability of the flight not being delayed when it does not rain is:
\[
P(\text{not delayed} \mid \text{not rain}) = \frac{P(\text{not rain and not delayed})}{P(\text{not rain})} = \frac{0.70}{0.82}
\]

Now, to solve this calculation and use Bayes' theorem, I will compute the values.The probability that the flight is not delayed given that it does not rain is approximately \( 0.854 \).

### Step 3: Calculate the probability of the flight being delayed, \( P(\text{delayed}) \)

The total probability of the flight being delayed, \( P(\text{delayed}) \), can be found by considering two scenarios: when it rains and when it does not rain. Using the law of total probability:

\[
P(\text{delayed}) = P(\text{delayed} \mid \text{rain}) \cdot P(\text{rain}) + P(\text{delayed} \mid \text{not rain}) \cdot P(\text{not rain})
\]

Since \( P(\text{not delayed} \mid \text{not rain}) \) is \( 0.854 \), the probability of the flight being delayed when it does not rain is:

\[
P(\text{delayed} \mid \text{not rain}) = 1 - 0.854 = 0.146
\]

Now we need \( P(\text{delayed} \mid \text{rain}) \). Since \( P(\text{delayed}) \) was given as 0.13, we can calculate the next steps accordingly using the law of total probability and plug it into Bayes' Theorem.

Let's compute the final conditional probability.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can guide you through solving the remaining steps manually or help with another problem.

Would you like me to walk you through it, or do you have any other questions?

Here are 5 follow-up questions you might find helpful:
1. What is Bayes' theorem and how does it apply to conditional probability?
2. How can we interpret the law of total probability in real-world problems?
3. How do we differentiate between conditional and joint probability in a problem like this?
4. What strategies can be used to break down multi-step probability problems?
5. How does complement rule help simplify complex probability calculations?

**Tip:** In probability, always check if events are independent or dependent, as this significantly affects how probabilities are calculated!

At LaGuardia Airport for a certain nightly flight, the probability that it will rain is 0.18 and the probability that the flight will be delayed is 0.13. The probability that it will not rain and the flight will leave on time is 0.7. What is the probability that it is raining if the flight has been delayed? Round your answer to the nearest thousandth.

Solve a Bayes' Theorem problem where we calculate the probability of rain given that a flight is delayed. With a detailed breakdown of conditional probability and key formulas, this solution is perfect for students in Grades 10-12.

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