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Let's solve **Question 16** from the uploaded image:

**Question 16:**
> There is a 43% chance that Riya walks to work. Given that Riya walks to work, she meets her friend Jasmine 30% of the time, and if she does not walk to work, she meets Jasmine 25% of the time. Find the probability that Riya walks to work on any particular day.

### Solution

Let's denote the following events:
- \( W \): Riya walks to work.
- \( M \): Riya meets Jasmine.

From the problem statement, we know:
- \( P(W) = 0.43 \), the probability that Riya walks to work.
- \( P(M | W) = 0.30 \), the probability that Riya meets Jasmine given that she walks to work.
- \( P(M | W^c) = 0.25 \), the probability that Riya meets Jasmine given that she does **not** walk to work.

We need to find the probability that Riya walks to work, \( P(W) \), which is already given as 43% or \( 0.43 \). 

However, if the question is asking for the probability that Riya walks to work on a particular day given that she met Jasmine, we would apply **Bayes' theorem**.

### Finding \( P(W | M) \)

According to Bayes' theorem:
\[
P(W | M) = \frac{P(M | W) \cdot P(W)}{P(M)}
\]

#### Step 1: Calculate \( P(M) \)

Using the law of total probability:
\[
P(M) = P(M | W) \cdot P(W) + P(M | W^c) \cdot P(W^c)
\]
where \( P(W^c) = 1 - P(W) = 1 - 0.43 = 0.57 \).

Substitute the known values:
\[
P(M) = (0.30)(0.43) + (0.25)(0.57)
\]
Calculate each term:
1. \( (0.30)(0.43) = 0.129 \)
2. \( (0.25)(0.57) = 0.1425 \)

So,
\[
P(M) = 0.129 + 0.1425 = 0.2715
\]

#### Step 2: Substitute into Bayes' formula

Now we can find \( P(W | M) \):
\[
P(W | M) = \frac{(0.30)(0.43)}{0.2715}
\]
\[
P(W | M) = \frac{0.129}{0.2715} \approx 0.4751
\]

### Final Answer
The probability that Riya walked to work on a particular day, given that she met Jasmine, is approximately **0.4751** or **47.51%**.

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### Related Questions
1. How would the probability change if the chance of meeting Jasmine without walking was increased?
2. What would \( P(W | M^c) \) be, where \( M^c \) is the event that Riya does not meet Jasmine?
3. How does Bayes' theorem apply in scenarios with conditional probabilities?
4. If \( P(W) \) changed to 50%, how would that affect \( P(W | M) \)?
5. How can we apply the law of total probability in real-life situations?

### Tip
When dealing with conditional probability questions, always check if Bayes' theorem is applicable by identifying "given" conditions.

There is a 43% chance that Riya walks to work. Given that Riya walks to work, she meets her friend Jasmine 30% of the time, and if she does not walk to work, she meets Jasmine 25% of the time. Find the probability that Riya walks to work on any particular day, given that she met Jasmine.

Learn how to calculate the probability that Riya walks to work given that she met her friend Jasmine using Bayes' Theorem and conditional probability. This problem covers key probability concepts, including Bayes' Theorem and the law of total probability. Ideal for high school students in Grades 10-12.

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