A survey shows that 70% of the college students have seen the advertisement of a new brand of drinks on television. The probability of a college student who has seen the advertisement buying the new brand of drinks is 0.85 while the probability of a college student who has not seen the advertisement buying the new brand of drink is 0.3.

Construct a probability tree diagram to illustrate the events.

Let's break this problem into events and probabilities to set up the probability tree diagram.

### Step 1: Define the Events
- Let \( A \) represent the event that a college student has seen the advertisement.
- Let \( \bar{A} \) represent the event that a college student has **not** seen the advertisement.
- Let \( B \) represent the event that a college student buys the new brand of drinks.
- Let \( \bar{B} \) represent the event that a college student does **not** buy the new brand of drinks.

### Step 2: Probabilities
From the problem, we know:
- \( P(A) = 0.7 \) (Probability that a student has seen the advertisement)
- \( P(\bar{A}) = 0.3 \) (Probability that a student has not seen the advertisement)

For the conditional probabilities:
- \( P(B|A) = 0.85 \) (Probability that a student buys the drink given that they saw the advertisement)
- \( P(B|\bar{A}) = 0.3 \) (Probability that a student buys the drink given that they did not see the advertisement)

We can also calculate the complementary probabilities:
- \( P(\bar{B}|A) = 1 - P(B|A) = 1 - 0.85 = 0.15 \)
- \( P(\bar{B}|\bar{A}) = 1 - P(B|\bar{A}) = 1 - 0.3 = 0.7 \)

### Step 3: Construct the Probability Tree

The tree starts with the first decision point: whether a student has seen the advertisement or not. Then, for each branch, it shows the probability of buying or not buying the drink.

```
                            Start
                              |
              -------------------------------------------------
              |                                               |
        Seen Ad (A)                                      Not Seen Ad (¬A)
         P(A) = 0.7                                        P(¬A) = 0.3
              |                                               |
       -----------------                               -----------------
       |               |                               |               |
    Buy (B)       Not Buy (¬B)                    Buy (B)         Not Buy (¬B)
   P(B|A) = 0.85  P(¬B|A) = 0.15                P(B|¬A) = 0.3   P(¬B|¬A) = 0.7
```

### Step 4: Multiply Probabilities for Each Branch
To calculate the joint probabilities for each path in the tree:
- \( P(A \cap B) = P(A) \times P(B|A) = 0.7 \times 0.85 = 0.595 \)
- \( P(A \cap \bar{B}) = P(A) \times P(\bar{B}|A) = 0.7 \times 0.15 = 0.105 \)
- \( P(\bar{A} \cap B) = P(\bar{A}) \times P(B|\bar{A}) = 0.3 \times 0.3 = 0.09 \)
- \( P(\bar{A} \cap \bar{B}) = P(\bar{A}) \times P(\bar{B}|\bar{A}) = 0.3 \times 0.7 = 0.21 \)

This probability tree diagram helps visualize the possible outcomes and their associated probabilities.

---

Would you like more details on the steps or how to interpret the tree diagram?

Here are 5 questions to further explore:
1. How do you compute the total probability that a student buys the drink?
2. What is the probability that a student does **not** buy the drink?
3. Can you use Bayes' Theorem to find the probability a student saw the ad given they bought the drink?
4. How would you modify the tree if the ad's effectiveness increased?
5. What if we included a third option like online ads? How would that affect the tree?

**Tip:** When working with probability trees, remember that each branch should sum to 1, so always check for completeness in both primary and conditional probabilities.

A survey shows that 70% of the college students have seen the advertisement of a new brand of drinks on television. The probability of a college student who has seen the advertisement buying the new brand of drinks is 0.85 while the probability of a college student who has not seen the advertisement buying the new brand of drink is 0.3. Construct a probability tree diagram to illustrate the events.

Explore the probability tree diagram illustrating the likelihood of college students buying a new brand of drinks based on a survey. Learn how conditional probabilities and joint probabilities are used to solve this real-world problem. Suitable for high school students learning probability concepts.

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