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Let's work through the probabilities step by step based on the provided table:

- **P(M)** is the probability that a study participant drank milk before bed. This is the total number of participants who drank milk divided by the total number of participants.

- **P(S)** is the probability that a study participant slept at least 7 hours. This is the number of participants who slept at least 7 hours divided by the total number of participants.

- **P(S|M)** is the conditional probability that a study participant slept at least 7 hours given that they drank milk. This is calculated by dividing the number of participants who drank milk and slept at least 7 hours by the total number of participants who drank milk.

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

- The total number of participants who drank milk = \( 480 \).
- The total number of participants = \( 1000 \).

So, 
\[
P(M) = \frac{480}{1000} = 0.48
\]

### Step 2: Calculate \( P(S) \)

- The number of participants who slept at least 7 hours = \( 550 \).
- The total number of participants = \( 1000 \).

So,
\[
P(S) = \frac{550}{1000} = 0.55
\]

### Step 3: Calculate \( P(S|M) \)

- The number of participants who drank milk and slept at least 7 hours = \( 264 \).
- The total number of participants who drank milk = \( 480 \).

So,
\[
P(S|M) = \frac{264}{480} = 0.55
\]

### Summary of Results:
- \( P(M) = 0.48 \)
- \( P(S) = 0.55 \)
- \( P(S|M) = 0.55 \)

Would you like more details or clarifications on any of these steps? Here are some related questions to think about:

1. What is the complement of \( P(S) \), the probability of sleeping less than 7 hours?
2. How would you calculate \( P(M \cap S) \), the joint probability of drinking milk and sleeping at least 7 hours?
3. How would you interpret the result of \( P(S|M) \)?
4. What is the probability that a participant did not drink milk but still slept at least 7 hours?
5. How could you use these probabilities to determine if drinking milk is correlated with sleeping at least 7 hours?

Tip: Conditional probabilities are useful for determining the likelihood of an event given that another event has already occurred!

A medical researcher is interested in the relationship between drinking warm milk before bed and sleeping at least 7 hours. Based on the given data, calculate the following probabilities: P(M), P(S), and P(S|M).

Explore how to calculate probabilities related to drinking milk before bed and sleeping at least 7 hours. This problem uses concepts of probability and conditional probability, including formulas for P(M), P(S), and P(S|M). Suitable for students in Grades 9-12.

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