At basketball tryouts, Jeremiah will shoot a \[1\]-point shot, a \[2\]-point shot, and a \[3\]-point shot one after the other. The table below shows Jeremiah's probability of making each shot:

ShotProbability of making

\[1\]-point

\[80\%\]

\[2\]-point

\[50\%\]

\[3\]-point

\[30\%\]

Assume the outcome of one shot doesn't change the probability of other shots.

The coach will record the total points Jeremiah scores from these \[3\] shots.

**Which graph represents the theoretical probability distribution of Jeremiah's total points?**

Choose 1 answer:

Choose 1 answer:

(Choice A)

A

(Choice B)

B

(Choice C)

C

To solve this problem, we need to calculate the theoretical probability distribution of the total points Jeremiah can score. Let's break it down step by step:

### Step 1: Possible Total Scores
Jeremiah can score a total of 0, 1, 2, 3, 4, 5, or 6 points from these three shots. These correspond to all possible combinations of making or missing each of the 1-point, 2-point, and 3-point shots.

### Step 2: Calculate Probabilities for Each Score

1. **0 points**: Miss all shots.
   - Probability = \((0.2) \times (0.5) \times (0.7) = 0.07\)

2. **1 point**: Make the 1-point shot and miss the other two.
   - Probability = \((0.8) \times (0.5) \times (0.7) = 0.28\)

3. **2 points**: Make the 2-point shot and miss the other two.
   - Probability = \((0.2) \times (0.5) \times (0.7) = 0.07\)

4. **3 points**: Make the 3-point shot and miss the other two.
   - Probability = \((0.2) \times (0.5) \times (0.3) = 0.03\)

5. **4 points**: Make the 1-point and 3-point shots but miss the 2-point shot.
   - Probability = \((0.8) \times (0.5) \times (0.3) = 0.12\)

6. **5 points**: Make the 1-point and 2-point shots but miss the 3-point shot.
   - Probability = \((0.8) \times (0.5) \times (0.7) = 0.28\)

7. **6 points**: Make all three shots.
   - Probability = \((0.8) \times (0.5) \times (0.3) = 0.12\)

### Step 3: Construct the Distribution
From these probabilities, we can build the probability distribution of total points:

- **0 points**: 0.07
- **1 point**: 0.28
- **2 points**: 0.07
- **3 points**: 0.03
- **4 points**: 0.12
- **5 points**: 0.28
- **6 points**: 0.12

### Step 4: Identify the Correct Graph
The correct graph should represent the probabilities calculated above, showing the distribution of total points Jeremiah can score.

Given that I can't see the graphs you have to choose from, use the distribution we've calculated:

- Highest probability at 1 and 5 points (0.28 each).
- Lower probabilities at 0, 2, 4, and 6 points.
- Very low probability at 3 points.

Choose the graph that most closely matches this probability distribution.

Would you like any more details or have further questions?

Here are 5 related questions to expand your understanding:
1. How would the probability distribution change if Jeremiah's probability of making the 3-point shot increased?
2. What would be the expected value of the total points Jeremiah would score?
3. How would the distribution change if the 1-point shot had a different probability?
4. What if Jeremiah shot a fourth shot with a different point value and probability?
5. How could you simulate this scenario using a computer program?

**Tip:** Understanding how to build a probability distribution from independent events is crucial in probability theory and can help in many real-world applications.

At basketball tryouts, Jeremiah will shoot a 1-point shot, a 2-point shot, and a 3-point shot one after the other. The table below shows Jeremiah's probability of making each shot:

Shot	Probability of making
1-point	80%
2-point	50%
3-point	30%

Assume the outcome of one shot doesn't change the probability of other shots.

The coach will record the total points Jeremiah scores from these 3 shots.

Which graph represents the theoretical probability distribution of Jeremiah's total points?

Choose 1 answer:
(Choice A)
A
(Choice B)
B
(Choice C)
C

Explore the theoretical probability distribution of Jeremiah's total points at basketball tryouts, based on his shooting probabilities for 1-point, 2-point, and 3-point shots. Understand how to calculate probabilities for independent events and analyze the distribution of outcomes.

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