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%.

Would you like a breakdown of any step or more details?

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.