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The problem asks to determine if the events "Makes 1st Shot" and "Makes 2nd Shot" are independent by checking if \( P(A \text{ and } B) = P(A) \cdot P(B) \), where:

- \( A \) = event of making the 1st shot,
- \( B \) = event of making the 2nd shot.

### Step 1: Find \( P(A) \)
The probability of making the 1st shot, \( P(A) \), is the number of times the player made the first shot divided by the total number of trials.

From the table:
- The total number of trials = 80,
- The number of times the 1st shot was made = 60.

So, \( P(A) = \frac{60}{80} = 0.75 \).

### Step 2: Find \( P(B) \)
The probability of making the 2nd shot, \( P(B) \), is the number of times the player made the second shot divided by the total number of trials.

From the table:
- The number of times the 2nd shot was made = 67.

So, \( P(B) = \frac{67}{80} = 0.8375 \).

### Step 3: Find \( P(A \text{ and } B) \)
The probability of making both the 1st and 2nd shots, \( P(A \text{ and } B) \), is the number of times the player made both shots divided by the total number of trials.

From the table:
- The number of times both the 1st and 2nd shots were made = 50.

So, \( P(A \text{ and } B) = \frac{50}{80} = 0.625 \).

### Step 4: Check for Independence
Now, check if \( P(A \text{ and } B) = P(A) \cdot P(B) \).

- \( P(A) \cdot P(B) = 0.75 \times 0.8375 = 0.628125 \).

Since \( P(A \text{ and } B) = 0.625 \) and \( P(A) \cdot P(B) = 0.628125 \), these two values are not equal. Therefore, the events "Makes 1st Shot" and "Makes 2nd Shot" are **not independent**.

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

1. How do dependent and independent events differ in probability theory?
2. What factors could cause two events to be dependent?
3. Can you explain conditional probability and how it relates to independence?
4. How can we use probability trees to visualize dependent events?
5. How would the analysis change if the probabilities were exactly equal?

**Tip**: Independence means that the outcome of one event does not affect the outcome of the other. Always check the multiplication rule to verify this.

Determine if the events 'Makes 1st Shot' and 'Makes 2nd Shot' are independent by checking the equation P(A and B) = P(A) * P(B). Show your work and reasoning!

This problem involves determining if making two basketball shots are independent events by using the equation P(A and B) = P(A) * P(B). It includes detailed steps for calculating probabilities of each shot and checking for independence.

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